Journal of Lie Theory Volume 11 (2001) 559–604 C ◯2001 Heldermann Verlag

On Asymptotic Behavior and Rectangular Band Structures in SL(2,ℝ)

Brigitte E. Breckner^{∗} and Wolfgang A.F. Ruppert

Communicated by K. H. Hofmann

Abstract. We associate with every subsemigroup of Sl(2, ℝ), not contained in a single Borel group, an ‘asymptotic object,’ a rectangular band which is defined on a closed subset of a torus surface. Using this concept we show that the horizon set (in the sense of Lawson [10]) of a connected open subsemigroup of Sl(2, ℝ) is always convex, in fact the interior of a three dimensional Lie semialgebra. Other applications include the classification of all exponential subsemigroups of Sl(2, ℝ) and the asymptotics of semigroups of integer matrices in Sl(2, ℝ).

1991 Mathematics Subject Classification: 22E15, 22E67, 22E46, 22A15, 22A25 Key words and phrases: asymptotic objects, asymptotic property, subsemigroups of Sl(2, ℝ), Lie semigroups, Lie semialgebras and their classification, compression semigroups, diamond product, rectangular domain, umbrella sets, Control Theory in Lie groups, asymptotics of integer matrix semigroups, semialgebraic sets.

1. Introduction

During the last two decades Lie semigroups and Lie wedges have been the central objects of interest in the structure theory of subsemigroups of Lie groups. With the Lie theory of groups as a model and some control theoretic ideas as additional input, a useful technical apparatus has been developed, allowing the passage from infinitesimal to local and global objects, and, conversely, from local or global Lie semigroups to their infinitesimal objects, which are Lie wedges. This machinery provided us with fairly detailed information about interesting types of Lie semigroups and about the interplay between infinitesimal, local and global properties of semigroups. A substantial part of the theory has been published in books and monographs, we only mention [5], [6], [14] and [8].

For some questions, however, these well established tools have to be complemented by definitely non-local concepts and methods. Most prominent among such problems is the determination and structural description of maximal

ISSN 0949–5932 / $2.50 ^{_{C} ◯} Heldermann Verlag subsemigroups in a connected Lie group. Let S be a maximal subsemigroup with nonempty interior of a connected Lie group G. It has been shown by J.D. Lawson [9] that if G is solvable then S must be a half space subsemigroup bounded by a closed subgroup of codimension one, so in this case S is a Lie semigroup. (A similar result holds if the radical of G is cocompact, cf. [6].) But if G is semisimple then S no longer needs to be a Lie semigroup. In fact, by the work of L.A.B. San Martin and P.A. Tonelli (cf., e.g., [18] and [19]), we know how to construct, if G is noncompact semisimple and has finite center, maximal closed subsemigroups S with nonempty interior which do not contain the identity. These semigroups appear as compression semigroups of certain subsets of a flag manifold G/P. In this theory an important question is whether a subsemigroup S of G meets a certain open subset of G, the control set of the action. This is a definitely non-local problem, which, as D. Mittenhuber ([12],[13]) has pointed out, is closely related with the horizon concept of J.D. Lawson in [10].

∗ This author was supported by the Deutsche Forschungsgemeinschaft.

The horizon of a subsemigroup S with nonempty interior of a Lie group G is defined as the set of all vectors in the Lie algebra for which the corresponding one parameter subsemigroup eventually stays in the interior of S. Obviously, the horizon of S is closed under multiplication with positive scalars, and if two elements in the horizon commute then their sum also lies in the horizon. So far further information about the structure of the horizon seems to be scarce, except in a few special cases.

If the subsemigroup S of G neither does cluster at the identity nor has interior points (in particular, if S is discrete) then there is usually little chance that the infinitesimal or local theory of subsemigroups of Lie groups will yield any structural information about S. But, even for S discrete we are faced with the natural analytic question asking for the behavior of S ‘at large.’

All these situations call for the study of asymptotic properties and asymptotic objects, alongside and in conjunction with infinitesimal properties and tangent objects. The present paper is intended to produce evidence for the feasibility of such a theory, offering a fairly comprehensive treatment of asymptotic properties in the special case of subsemigroups of Sl(2, ℝ). We shall show that, at least in this special case,

• workable definitions of asymptotic objects, associated with not necessarily connected subsemigroups, can be given,

• these asymptotic objects bear a ‘rectangular band’ multiplicative structure, which is obtained easily from the multiplication of S by a limiting process,

• the so defined asymptotic objects can be explicitly computed from the Lie wedge L(S) if S is a Lie semigroup (thus infinitesimal and asymptotic structures supplement each other nicely),

• the asymptotic objects are very helpful for the determination and structural description of the horizon sets of Lawson and similar objects (‘umbrella sets’, defined below), they can be used also for the determination of the conjugacy classes of exponential subsemigroups of Sl(2, ℝ).

Perhaps the most surprising among the results of this paper is the fact that for a connected open subsemigroup S ≠ Sl(2, ℝ) the umbrella set (which in this case coincides with Lawson’s horizon set)

Umb(S) = {X ∈ 𝔰𝔩(2, ℝ) | exp ℝ^{+}X ∩ S ≠ Ø}

is an open convex cone whose closure is the Lie wedge of an exponential subsemigroup of Sl(2, ℝ). This feature certainly does not hold in general for connected open subsemigroups of Lie groups.

In a subsequent paper ([17]) we shall exploit the present concepts and the ensuing theory for the explicit construction of Bohr compactifications of subsemigroups of Sl(2, ℝ).

We temporarily postpone, however, the further extension of the theory to higher dimensions. Many important aspects of the structure of Sl(2, ℝ) are no longer present in a more general context, and we cannot hope to develop a comparably satisfying theory by straightforward generalization. For instance, as mentioned above, in Sl(2, ℝ) a central role is played by the exponential subsemigroups with nonvoid interior, whereas for noncompact simple Lie groups of dimension > 3 no exponential subsemigroups with interior points exist. The group Sl(2, ℝ) has a unique standing among simple Lie groups (similar to the unique role of ℝ among abelian Lie groups) and it is also the aim of this paper to further elucidate its special structure.

Acknowledgements. We want to thank especially Karl Heinrich Hofmann, who evoked the interest of both authors in questions which eventually led to investigations some of which are presented here, and who always has taken an active interest in these studies. Special thanks go also to the Deutsche Forschungsgemeinschaft (DFG) for giving financial support to the first author during her stay at Darmstadt and at Vienna while working on this project.

2. Setting the stage

2.1. Notation. Throughout these notes we use the symbol ℝ^{+} for the set of strictly positive reals and ℝ_{0}^{+} for the set of nonnegative reals, we write ℕ for the positive integers and ℕ_{0} for ℕ ∪ {0}. The interior of a subset A of a topological space is denoted by Int A, the closure of A by \overline{A}.

We denote the Killing form of 𝔰𝔩(2, ℝ) by Kill and write

Kill^{+} =^{def} \{X ∈ 𝔰𝔩(2, ℝ) | − det(X) = \frac{1}{8}Kill(X, X) > 0\}.

For any X ∈ Kill^{+} we let

Δ(X) =^{def} \sqrt{− det(M)}.

Similarly, Kill^{0} [Kill^{−}] will denote the set of all X ∈ 𝔰𝔩(2, ℝ) with Kill(X, X) = 0 [Kill(X, X) < 0]. The set Kill^{0} will be also called the light cone.

Furthermore, in accordance with the notation of [5]:

H = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) , P = ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) , Q = ( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} ) .

Note that the Borel algebras in 𝔰𝔩(2, ℝ) are exactly the conjugates of ℝH + ℝP.

2.2. The image of the exponential function.

(i) \overline{exp(Kill^{+})} = exp(\overline{Kill^{+}}) and the restriction of exp to \overline{Kill^{+}} is a diffeomorphism.

(ii) Sl(2, ℝ) = exp(𝔰𝔩(2, ℝ)) ∪ (−1) · \overline{exp(Kill^{+})} and \overline{exp(Kill^{+})} ∩ (−1) · \overline{exp(Kill^{+})} = Ø.

Proof. These assertions are long known, they can be inferred, e.g., from the discussion in [5], p. 416ff.

Notation. In view of 2.2(i) we can extend the usual log-function exp(B) → B, where B is a Campbell-Hausdorff neighborhood, to a map

exp(B) ∪ exp(\overline{Kill^{+}}) → B ∪ \overline{Kill^{+}}.

This extension is unique, we also denote it by log.

2.3. Characterizations involving the trace. Let g = \binom{a b}{c d} be an element in Sl(2, ℝ). Then the following assertions hold:

(i) g ∈ exp(𝔰𝔩(2, ℝ)) if and only if either trace(g) > −2 or g = −1.

(ii) −2 < trace(g) if and only if g lies in the interior of exp(𝔰𝔩(2, ℝ)).

(iii) −2 < trace(g) < 2 if and only if g is conjugate to a rotation x ∈ SO(2, ℝ) with x ≠ 1, −1.

(iv) trace(g) > 2 if and only if g lies in exp(Kill^{+}).

(v) trace(g) = 2 if and only if g is unipotent, i.e., g ∈ exp(Kill^{0}).

(vi) trace(g) = 2 cos(qπ) ∉ {−2, 2} for some nonzero rational q if and only if g has finite order > 2.

Proof. In view of 2.2 and by inspection these assertions follow readily if g is either an upper triangular matrix or a rotation matrix. Since every matrix in Sl(2, ℝ) is conjugate to either an upper triangular or a rotation matrix, and since the trace is invariant under conjugation, this implies the assertions.

2.4. The reduced logarithm. For technical convenience we introduce the reduced logarithm as the map

rlog: exp(Kill^{+}) → Δ^{−1}(1), g ↦ \frac{1}{Δ(log(g))} · log(g).

2.5. Explicit formulas for exp and rlog. Let g = \binom{a b}{c d}.

(i) If trace(g) > 2 then

rlog(g) = ( \frac{2c}{u} −\frac{a−d}{u}^{\frac{a−d}{u} \frac{2b}{u}} ) with u = \sqrt{(a + d)^{2} − 4}.

Furthermore

log(g) = log(\frac{a + d + u}{2}) · rlog(g).

(ii) Conversely, for any real numbers α, β, γ with α^{2} + βγ = 1 and any nonzero t ∈ ℝ we have

exp t · ( \begin{matrix}α & β \\ γ & −α \end{matrix} ) = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ exp(Kill^{+}),

where

a = cosh(t) + α sinh(t) = \frac{1 + α}{2}e^{t} + \frac{1 − α}{2}e^{−t},  b = sinh(t)β,
c = sinh(t)γ,  d = cosh(t) − α sinh(t) = \frac{1 − α}{2}e^{t} + \frac{1 + α}{2}e^{−t}.

Proof. The formulas in (i),(ii) can be checked quickly by straightforward computation; they are just variants of the formulas in [5], p.417.

2.6. Remark. If X ∈ 𝔰𝔩(2, ℝ) is nilpotent then X^{2} = 0 and exp tX = 1 + tX. Conversely, if g ∈ Sl(2, ℝ) is unipotent then log(g) = g − 1.

2.7. Lie wedges and Lie semigroups. (cf. [6], p.19ff) Recall that if S is a closed subsemigroup with 1 ∈ S of a Lie group G with Lie algebra 𝔤 then the Lie wedge L(S) of S is the set of all X ∈ 𝔤 with exp(ℝ_{0}^{+}X) ⊆ S. The Lie wedge L(S) is indeed a wedge, i.e., a closed convex set which is closed under addition. Furthermore, for every X ∈ L(S) ∩ −L(S) we have e^{adX}L(S) = L(S).

If S is the smallest closed subsemigroup generated by exp(L(S)) then S is called a Lie semigroup and W = L(S) is said to be a global Lie wedge with respect to G.

2.8. Lie semialgebras. (cf. [5], p.86.) Let W be a wedge in a Lie algebra 𝔤. Then W is called a Lie semialgebra if it is a local semigroup with respect to the Campbell-Hausdorff multiplication ∗, i.e., there exists a Campbell-Hausdorff neighborhood B such that (B ∩ W) ∗ (B ∩ W) ⊆ W.

In the Lie algebra 𝔰𝔩(2, ℝ) every Lie semialgebra with nonvoid interior is the intersection of half spaces bounded by a Borel algebra (cf. [5], II.3.7, p.109).

3. Basic properties of subsemigroups of Sl(2, ℝ)

3.1. This section is intended to provide a handy summary of those basic properties of subsemigroups of Sl(2, ℝ) which are used (or supposed to be known) in the sequel. Most of the results are from [5] and/or belong to the general folklore of the subject.

3.2. The interior of a subsemigroup in a topological group. Let S be a subsemigroup of a topological group G with nonvoid interior. Then the following assertions hold:

(i) The set Int S of interior points of S is an ideal in S.

(ii) If 1 ∈ \overline{Int S} then Int S is dense in S and Int(\overline{S}) = Int S.

(iii) Suppose that G is locally connected and that S is connected. If 1 ∈ \overline{S} then Int S is connected.

Proof. Assertions (i), (ii) form Proposition V.0.15 of [5], p.370.

(iii) Pick a, b ∈ Int S and let U, V be connected open sets with a ∈ U ⊆ Int S, b ∈ V ⊆ Int S. Since 1 ∈ \overline{S} the intersections U ∩ aS and V ∩ Sb are nonvoid, so U ∪ aS and V ∪ Sb are connected subsets of Int S. Since aS ∩ Sb contains the nonvoid subset aSb we therefore conclude that the union U ∪ aS ∪ V ∪ Sb is a connected subset of Int S containing both a and b. This implies (iii).

3.3. Connected components of a topological semigroup. (cf.[16] Proposition 3.1) Let S be a topological semigroup. For x ∈ S we denote with C(x) the connected component of S which contains x.

(i) For every pair x, y ∈ S we have C(x)C(y) ⊆ C(xy) (note that C(x)C(y) is connected and contains xy). In other words: the relation x ∼_{C} y ⟺ C(x) = C(y) is a congruence on S.

(ii) x^{2} ∈ C(x) if and only if C(x) is a semigroup.

(iii) For every x ∈ S the union S_{c}(x) = ⋃_{k∈ℕ} C(x^{k}) is a subsemigroup of S.

3.4. Proposition. Let S be a subsemigroup of Sl(2, ℝ) with nonvoid interior Int S in Sl(2, ℝ) and suppose that S ≠ Sl(2, ℝ). Then the following assertions hold:

(i) Int S ⊆ exp(Kill^{+})∪(−1)·exp(Kill^{+}) and Int S ∩exp(Kill^{+}) is nonempty.

(ii) The closure of S cannot contain a one dimensional torus group.

(iii) If S is connected then S ⊆ \overline{exp(Kill^{+})} and Int S ⊆ exp(Kill^{+}).

(iv) If 1 ∈ \overline{Int S} then S_{1} =^{def} \overline{Int S ∩ exp(𝔰𝔩(2, ℝ))} and its interior Int(S_{1}) = Int S ∩ exp(Kill^{+}) are connected semigroups.

Proof. (i) Since Int S ≠ Sl(2, ℝ) and since every neighborhood of a compact element contains an element of finite order we conclude that Int S cannot contain compact elements. Similarly there are compact elements, and hence elements of finite order, in every vicinity of a unipotent element, so Int S cannot contain unipotent elements. Thus by the discussion in 2.3 above Int S ∩ exp(𝔰𝔩(2, ℝ)) ⊆ exp(Kill^{+}) and the first assertion follows. If s = −g ∈ Int S with g ∈ exp(Kill^{+}) then s^{2} = g^{2} ∈ Int S ∩ exp(Kill^{+}), so Int S ∩ exp(Kill^{+}) cannot be empty.

(ii) Suppose that, contrary to our assertion, \overline{S} contains a one dimensional torus subgroup T. Pick an inner point x ∈ Int S. But trace(−x) = − trace(x) and T is connected, so we conclude that there is a t ∈ T with trace(tx) = 0. For all s ∈ S we have sx ∈ Int S and for s sufficiently near to t we also have trace(sx) ∈] − 1, 1[, which means that the interior of S contains a compact element (2.3(iii)), a contradiction.

(iii) Since S is connected and contains inner points, the set trace(S) is a nondegenerate interval. If there were a point g ∈ S \ \overline{exp(Kill^{+})} then by 2.3(iv) trace(g) < 2. On the other hand, by (i), there is an element h ∈ S ∩ exp(Kill^{+}), and trace(h) > 2, by 2.3(iv). But S is connected, so trace(S) would be a neighborhood of 2 and therefore we could find a k ∈ S such that trace(k) = 2 cos(α), with α rationally independent of π. Since the powers k^{n} of such an element are dense in a one dimensional torus group, we arrive at a contradiction to (ii). This implies S ⊆ \overline{exp(Kill^{+})}. The remaining inclusion follows from 2.3(iv) and (v).

(iv) Suppose that 1 ∈ \overline{Int S}. Then Int S is dense in S. We assume, without losing generality, that S is open. It was shown in [16] that in PSl(2, ℝ) = Sl(2, ℝ)/{1, −1} every open subsemigroup clustering at 1 is connected. Let q: Sl(2, ℝ) → PSl(2, ℝ) = Sl(2, ℝ)/{1, −1} be the natural quotient morphism. Then q(S) is open and clusters at 1, hence is connected. Its inverse image q^{−1}q(S) = S ∪ (−1)S is an open semigroup and has exactly two connected components, namely (S ∪ −S) ∩ exp(Kill^{+}) and (S ∪ −S) ∩ − exp(Kill^{+}). The connected component (S ∪ −S) ∩ exp(Kill^{+}) is a semigroup (by 3.3(ii)) and contains S ∩ exp(Kill^{+}). Thus the semigroup generated by S ∩ exp(Kill^{+}) is contained in exp(Kill^{+}), so S ∩ exp(Kill^{+}) is a semigroup. The restriction of q to S ∩ exp(Kill^{+}) is an isomorphic embedding into PSl(2, ℝ). Since the open semigroup q(S ∩ exp(Kill^{+})) clusters at 1, it is connected and therefore S ∩ exp(Kill^{+}) is connected. The assertion follows.

3.5. Proposition. Let S be an open subsemigroup of Sl(2, ℝ) with S ≠ Sl(2, ℝ). Then for every x ∈ S the following assertions hold:

(i) The subsemigroup S_{c}(x) = ⋃_{k∈ℕ} C(x^{k}) is open.

(ii) Suppose that x belongs to the exponential image in Sl(2, ℝ). Then S_{c}(x) is contained in exp(Kill^{+}) and the sequence ⟨C(x^{n})⟩ is eventually constant: there exists a natural number n_{0} such that C(x^{n}) = C(x^{n_{0}}) for n ≥ n_{0} . Moreover, C(x^{n_{0}}) is an ideal in S_{c}(x) and S_{c}(x)/ ∼_{C} is a finite nilpotent semigroup.

(iii) Suppose that x does not belong to the exponential image in Sl(2, ℝ). Then the powers of C(x) in the semigroup S/ ∼_{C} form a finite subsemigroup N(x) and the minimal ideal of N(x) is a cyclic group of order two.

Proof. Assertion (i) follows from the fact that in a locally connected space the connected components of open subsets are open.

(ii) If x ∈ exp(𝔰𝔩(2, ℝ)) then x ∈ exp(Kill^{+}) (by 3.4(i)) and x^{k} ∈ exp(Kill^{+}), for any k ∈ ℕ. Since C(x^{k}) is connected and open we see from 3.4(i) that C(x^{k}) ⊆ exp(Kill^{+}), for any k ∈ ℕ, so S_{c}(x) ⊆ exp(Kill^{+}).

Let x = exp X. Since C(x) is open we find a number n ∈ ℕ with y =^{def} exp((1 + \frac{1}{n})X) ∈ C(x). Then x^{n+1} = y^{n} ∈ C(x)^{n} ⊆ C(x^{n}). By connectedness, C(x)C(x^{n}) ∪ C(x^{n})C(x) ⊆ C(x^{n}). Also, we conclude by induction that C(x)^{n}C(x^{n}) ⊆ C(x^{n}), so the connected set C(x^{n})C(x^{n}) meets the component C(x^{n}) and therefore is contained in C(x^{n}). This finishes the proof of (ii).

(iii) The powers of x with odd exponents belong to − exp(Kill^{+}), those with even exponents to exp(Kill^{+}). Now (ii) implies the assertion.

3.6. Exponential subsemigroups of a Lie group. Recall that a closed subsemigroup S of a Lie group G is called exponential if it is the exponential image exp(W) of its Lie wedge W = L(S).

A wedge in the Lie algebra 𝔤 of G is called exponential with respect to G, or G-exponential, or exponential for short, if it is the Lie wedge of an exponential subsemigroup of G. Note that if G → G_{1} is a covering homomorphism (so that 𝔤 can be considered as the Lie algebra of G_{1} as well) and if W is an exponential wedge with respect to G then W is exponential also with respect to G_{1} — provided that W is global with respect to G_{1} .

3.7. The exponential subsemigroup Sl(2, ℝ)^{+} . (cf.[5], p. 419ff)

Following [5] we write Sl(2, ℝ)^{+} for the semigroup of matrices with nonnegative entries in Sl(2, ℝ). The corresponding Lie wedge

L(Sl(2, ℝ)^{+}) = ℝH + ℝ_{0}^{+}P + ℝ_{0}^{+}Q

is denoted by 𝔰𝔩(2, ℝ)^{+} . Obviously, 𝔰𝔩(2, ℝ)^{+} is a Lie semialgebra and Sl(2, ℝ)^{+} is an exponential subsemigroup of Sl(2, ℝ). It has been shown in [5] (p.421, V.4.30) that

(i) if a subsemigroup S of Sl(2, ℝ) contains Sl(2, ℝ)^{+} then its 1-component coincides with Sl(2, ℝ)^{+} and S ⊆ Sl(2, ℝ)^{+} ∪ (−1)Sl(2, ℝ)^{+} .

In particular, Sl(2, ℝ)^{+} is a maximal connected subsemigroup of Sl(2, ℝ). If W is the Lie wedge of a closed subsemigroup with nonvoid interior S ≠ Sl(2, ℝ) then W cannot meet Kill^{−} (by 3.4(ii)), hence for each of the convex halves of Kill^{−} there exists a support plane separating it from W. These two support planes dissect 𝔰𝔩(2, ℝ) into four open quadrants and W is contained in the closure W_{0} of one of the two quadrants not meeting Kill^{−}. The edge H(W_{0}) = W_{0} ∩ −W_{0} of W_{0} is a Cartan subalgebra, hence conjugate to ℝH. It follows that W_{0} must be conjugate to 𝔰𝔩(2, ℝ)^{+} . Thus

(ii) the Lie wedge of a Lie subsemigroup with nonvoid interior S ≠ Sl(2, ℝ) is always contained in a conjugate of 𝔰𝔩(2, ℝ)^{+} ,

(iii) every Lie subsemigroup with nonvoid interior S ≠ Sl(2, ℝ) is contained in a conjugate of Sl(2, ℝ)^{+} .

3.8. Lie wedges of three dimensional exponential subsemigroups. Let W be a wedge in 𝔰𝔩(2, ℝ), let S = exp(W) ⊆ Sl(2, ℝ) and assume that W does not lie in a subalgebra of dimension < 3. Then the following assertions are equivalent:

(i) S is a semigroup;

(ii) S is a closed semigroup;

(iii) W is the intersection of conjugates of 𝔰𝔩(2, ℝ)^{+} .

(iv) W is a Lie semialgebra contained in \overline{Kill^{+}} .

(v) W is an exponential wedge.

Proof. The implications (ii) ⟹ (i), (ii)⟺(v), and (iii) ⟹ (iv) are trivial.

(i) ⟹ (iv) We first note that S ≠ Sl(2, ℝ), since Sl(2, ℝ) is not exponential, and that Int S is nonvoid, since W is not contained in a two dimensional subalgebra of 𝔰𝔩(2, ℝ) ([5], Theorem V.1.10, p.377, and p.382). Also, S is connected, hence 3.4(iii) applies and shows that S is contained in \overline{exp(Kill^{+})}, so W = log(S) ⊆ \overline{Kill^{+}} . Since the restriction of exp to \overline{Kill^{+}} is a diffeomorphic embedding it follows that for any Campbell-Hausdorff neighborhood B

(B ∩ W) ∗ (B ∩ W) = log(exp((B ∩ W) ∗ (B ∩ W)))
= log(exp(B ∩ W) exp(B ∩ W))
⊆ log(exp(W)) = W,

so W is a semialgebra.

(iv) ⟹ (iii) By Theorem II.3.7 of [5] (p. 109) we know that W is the intersection of a family F of half space semialgebras each of which is bounded by a Borel algebra. Since W is a Lie semialgebra, and not contained in a Borel subalgebra, it has nonvoid interior. Let K_{1} , K_{2} be the two connected components of Kill^{−}. Then each half space semialgebra contains exactly one of either K_{1} or K_{2} . By 3.7(ii) there exist at least one pair W_{1}, W_{2} ∈ F with K_{1} ⊆ W_{1} , K_{2} ⊆ W_{2} , and for every such pair the intersection W_{1} ∩ W_{2} is conjugate to 𝔰𝔩(2, ℝ)^{+} and contains W. Clearly, W is the intersection of all such intersection wedges W_{1} ∩ W_{2} . Thus W is obtained as the intersection of conjugates of 𝔰𝔩(2, ℝ)^{+} .

(iii) ⟹ (ii) The restriction of exp to \overline{Kill^{+}} is a diffeomorphism and the exponential image of 𝔰𝔩(2, ℝ)^{+} is the closed subsemigroup Sl(2, ℝ)^{+} of all matrices in Sl(2, ℝ) with nonnegative entries. Thus we see that exp(W), being the intersection of conjugates of Sl(2, ℝ)^{+} , must be a closed subsemigroup of Sl(2, ℝ).

4. The diamond product

4.1. In the present notes we shall not discuss possible formal definitions for the terms ‘asymptotic property’ and ‘asymptotic object.’ Let us emphasize only that a natural requirement for an ‘asymptotic property’ seems to be: if S has this property then each of its two-sided ideals has this property. In the same vein, an ‘asymptotic object’ ought to be a functor assigning to certain semigroups S a structure A(S), so that A(S) = A(I) for every two-sided ideal I of S. Also, taking as a model the case of additive subsemigroups of ℝ^{n}, we would like to have the asymptotic objects carry good information about ‘asymptotic directions,’ i.e., limits of matrices X ∈ 𝔰𝔩(2, ℝ) with exp ℝ^{+}X ∩ S ≠ Ø.

The guiding idea of our subsequent definitions is to identify directions with matrices in 𝔰𝔩(2, ℝ) of length 1 and to define a special kind of multiplication of directions. Instead of introducing an arbitrary Euclidean distance we measure the length in terms of the Killing form, this is technically more convenient and better adapted to the structures we are interested in.

4.2. The fundamental hyperboloid. We henceforth call the hyperboloid of revolution

Hyp =^{def} Δ^{−1}(1) = {αH + βP + γQ | α^{2} + βγ = 1}

the fundamental hyperboloid. Note that the “light cone” Kill^{0} = {αH +βP +γQ | α^{2} + βγ = 0} is the asymptotic cone of Hyp.

For our purposes the point of interest is that the points of the fundamental hyperboloid correspond to the one parameter subsemigroups leading into exp(Kill^{+}). (These are exactly those one parameter subsemigroups which meet open subsemigroups S ≠ Sl(2, ℝ).) A drawback of working with Hyp is that it is not a compact surface, so we shall embed it later into a product of flag manifolds. The advantage of Hyp is that it allows easy calculations and that the multiplication introduced below has a convincing geometric interpretation.

4.3. Horizontal and vertical lines. For any X ∈ Hyp let 𝔭_{X}, 𝔮_{X} be the (one-dimensional) eigenspaces of ad X with, respectively, eigenvalue 2,-2. Then we define the horizontal line through X,

hor(X) =^{def} X + 𝔭_{X},

and the vertical line through X

vert(X) =^{def} X + 𝔮_{X}.

Both of these sets are contained in the fundamental hyperboloid Hyp.

4.4. Elementary Properties of horizontal and vertical lines. (i) For every X ∈ Hyp the convex span of hor(X) ∪ vert(X) is the tangent plane of Hyp at X and hor(X) ∩ vert(X) = {X}.

(ii) Because of 𝔭_{X} = 𝔮_{−X} we see that hor(X) = − vert(−X) and vert(X) = − hor(−X), so the lines hor(X) and vert(−X) are parallel to each other.

(iii) For later use we also note that for every Y ∈ hor(−X) ∪ vert(−X) the sum X + Y is nilpotent.

(iv) The linear span of hor(X) is the Borel algebra 𝔟_{1} = ℝX + 𝔭_{X} = ℝ · hor(X) ∪ 𝔭_{X}, the linear span of vert(X) is the Borel algebra 𝔟_{2} = ℝX + 𝔮_{X} = ℝ · vert(X) ∪ 𝔮_{X}. Note that 𝔟_{1} ∩ Hyp = hor(X) ∪ vert(−X) and 𝔟_{2} ∩ Hyp = vert(X) ∪ hor(−X).

(v) Distinct horizontal [vertical] lines do not intersect. (Indeed, Y ∈ hor(X) implies hor(Y) = hor(X), and Y ∈ vert(X) implies vert(X) = vert(Y).)

(vi) The horizontal line hor(X) intersects the vertical line vert(Y) if and only if Y ∉ − hor(X) (or, equivalently, if X ∉ − vert(Y)).

(vii) Under any automorphism of 𝔰𝔩(2, ℝ) horizontal lines go to horizontal lines and vertical lines go to vertical lines.

4.5. Explicit formulas for hor(X) and vert(X). For X = αH + βP + γQ ∈ Hyp we have hor(X) = X + ℝA, vert(X) = X + ℝB with

A = \{ \begin{matrix}β(α − 1)H + β^{2}P − (α − 1)^{2}Q & if (α, β) ≠ (1, 0); \\ −γH + 2P − \frac{γ^{2}}{2} Q & if (α, β) = (1, 0); \end{matrix}

B = \{ \begin{matrix}β(α + 1)H + β^{2}P − (α + 1)^{2}Q & if (α, β) ≠ (−1, 0); \\ γH + 2P − \frac{γ^{2}}{2} Q & if (α, β) = (−1, 0). \end{matrix}

We leave the straightforward proof to the reader.

4.6. Remarks about the orbit structure of Hyp.

(i) The set Hyp is the orbit of any of its points X under the adjoint action of Sl(2, ℝ) on 𝔰𝔩(2, ℝ). Thus, fixing X = H, we identify Hyp with the homogeneous space Sl(2, ℝ)/Z exp(ℝH), where Z = {I, −I} is the center of Sl(2, ℝ). (For X = P + Q the isotropy group is just SO(1, 1).)

(ii) The space Hor of all horizontal lines in Hyp (endowed with the Vietoris topology) can be identified with the flag manifold (the simplest possible) Sl(2, ℝ)/Z exp(ℝH + ℝP). This homogeneous space is homeomorphic with SO(2)/Z, which is also homeomorphic with the one dimensional torus 𝕋. The injection ℝH → ℝH + ℝP induces the natural map

Sl(2, ℝ)/Z exp(ℝH) → Sl(2, ℝ)/Z exp(ℝH + ℝP),

which to every Z exp(ℝH)-coset assigns the Z exp(ℝH + ℝP)-coset containing it; this mapping corresponds to the map hor: Hyp → Hor, X ↦ hor(X). Thus we see that hor is a continuous and open mapping.

(iii) Similarly, the space Vert of all vertical lines is identified with the homogeneous space Sl(2, ℝ)/Z exp(ℝH + ℝQ), which is homeomorphic with the one dimensional torus 𝕋, and we note that the map vert: Hyp → Vert, X ↦ vert(X), is a continuous and open mapping. It is convenient to write the quotient space Sl(2, ℝ)/Z exp(ℝH + ℝQ) as the set of left cosets Z exp(ℝH + ℝQ) · g, g ∈ Sl(2, ℝ).

(iv) We conclude from 4.4(vi) that vert(hor(X)) = Vert \{vert(−X)} and that hor(vert(X)) = Hor \{hor(−X)}.

(v) Further information about actions on Hyp and about its order structure can be found, e.g., in [6], p.57ff.

4.7. Proposition. We retain the notation of the preceding paragraph. The following assertions hold:

(i) The map c: Hyp → Hor × Vert, X ↦ (hor(X), vert(X)), is an open topological embedding of Hyp into the space Hor × Vert ≅ 𝕋 × 𝕋.

(ii) c maps Hyp onto the set c(Hyp) = {(h, v) ∈ Hor × Vert | h ≠ −v}, which is dense in Hor × Vert.

(iii) The inverse of the corestriction Hyp → c(Hyp) of c sends every pair (h, v) with h ≠ −v to the unique point X ∈ h ∩ v.

Remark. In terms of the identifications in 4.6 the map c is defined by

c(g · Z exp(ℝH)) = (g · Z exp(ℝH + ℝP), Z exp(ℝH + ℝQ) · g^{−1}).

Its inverse is given (under the proviso that h∩v ≠ Ø) by (h, v) ↦ g ·Z exp(ℝH), for any g ∈ h ∩ v.

Proof. (i) By 4.6(ii) and (iii) we know that Hor × Vert is homeomorphic with 𝕋 × 𝕋 and that the map c is continuous. By 4.4(i) c is injective, hence its restriction to any compact subset of Hyp is a topological embedding. It follows by the invariance of domains that c is an open map, hence an open topological embedding.

Assertions (ii) and (iii) follow from 4.4(vi).

4.8. The compactification (Hyp^{−}, c). The above proposition shows that the space Hor × Vert, together with the map c, is a compactification of Hyp. We abbreviate Hor × Vert by Hyp^{−}.

4.9. Nilpotency Points in Hyp^{−}. We henceforth call an element of the form (hor(X), − hor(X)) = (hor(X), vert(−X)), X ∈ Hyp, a nilpotency point.

(i) The set of all nilpotency points in Hyp^{−} is the remainder Hyp^{−} \c(Hyp) of the compactification (Hyp^{−}, c).

(ii) The set of all nilpotency points in Hyp^{−} can be identified with the space of all one dimensional subspaces ℝ·N, N ∈ Kill^{0} . Under this identification the nilpotency point (hor(X), − hor(X)) corresponds to the line hor(X) − X, for every X ∈ Hyp.

(iii) For every X ∈ Hyp the closure of c(hor(X)) [c(vert(X)] in Hyp^{−} contains exactly one nilpotency point, namely the point (hor(X), − hor(X)) [(− vert(X), vert(X))].

(iv) Let h ∈ Hor, v ∈ Vert. If (h, v) is not a nilpotency point then ℝh ∩ ℝv = ℝ(h ∩ v). If h = hor(X) and v = − hor(X) then ℝh = ℝv, moreover \overline{ℝh} = ℝh ∪ 𝔭_{X}.

4.10. Proposition. Let ⟨X_{n}⟩ be a sequence in Hyp such that for suitably chosen real numbers t_{n} the limit N = lim t_{n}X_{n} exists and is a nonzero nilpotent matrix with N ∈ hor(X)−X for some X ∈ Hyp. Then lim c(X_{n}) = (hor(X), − hor(X)).

Proof. We assume, with no loss of generality, that X = H, N = P. Write X_{n} = α_{n}H + β_{n}P + γ_{n}Q. Then

|t_{n}| = Δ(t_{n}X_{n}) → 0, t_{n}α_{n} → 0, t_{n}β_{n} → 1, t_{n}γ_{n} → 0.

In particular, |β_{n}| → ∞. Applying our formulas in 4.5 we see that for every s ∈ ℝ

\begin{matrix}X_{n}+ & s(β_{n}(α_{n} − 1)H + β_{n}^{2}P − (α_{n} − 1)^{2}Q) ∈ hor(X_{n}), \\ X_{n}+ & s(β_{n}(α_{n} + 1)H + β_{n}^{2}P − (α_{n} + 1)^{2}Q) ∈ vert(X_{n}). \end{matrix}

(Note that β_{n} ≠ 0 for sufficiently large n.) We now put s = −1/β_{n} and find

hor(X_{n}) = hor (H + \frac{2(1 − α_{n})}{β_{n}}Q) , vert(X_{n}) = vert (−H + \frac{2(1 + α_{n})}{β_{n}}Q) .

Next we choose n so large that t_{n} ≠ 0 and compute

lim_{n→∞} \frac{2(1 ∓ α_{n})}{β_{n}} = lim_{n} \frac{2(t_{n} ∓ t_{n}α_{n})}{t_{n}β_{n}} = 0

which implies that lim hor(X_{n}) = hor(H) and lim vert(X_{n}) = vert(−H). The assertion follows.

4.11. Remark. Note that by Proposition 4.10 the space Hyp^{−} can be considered as the quotient space of the space of all one parameter subsemigroups contained in exp(\overline{Kill^{+}}), where for each N ∈ Kill^{0} the one parameter semigroups exp(ℝ_{0}^{+}N) and exp(−ℝ_{0}^{+}N) are identified. Occasionally this identification requires some care when nilpotency points are involved.

4.12. Rectangular bands. (i) Recall that every cartesian product S = M ×N of nonvoid sets is an idempotent semigroup with respect to the multiplication (m, n)(m′, n′) = (m, n′). Such semigroups are called rectangular bands (cf., e.g.,

[3] I, p.25).

(ii) Note that with respect to this multiplication every subset M_{n} = M × {n}, n ∈ N, is a left zero semigroup, and every subset N_{m} = {m} × N, m ∈ M, is a right zero semigroup. Moreover, for fixed (m, n) ∈ S the map M_{n} × N_{m} → S, ((x, n), (m, y)) ↦ (x, n)(m, y) = (x, y), is an isomorphism.

(iii) Every rectangular band S is a simple semigroup, that is, if I is a two sided ideal in S then I = S.

4.13. Elementary properties of topological rectangular bands. Let M and N be two nonvoid topological spaces and let S = M × N be the associated rectangular band. Then the following assertions hold:

(i) S is a topological semigroup, i.e., the multiplication is jointly continuous. Moreover, the multiplication of S is an open mapping.

(ii) A subset S′ of S is a subsemigroup [open subsemigroup] [closed subsemigroup] [connected subsemigroup] if and only if there exist nonvoid [nonvoid open] [nonvoid closed] [nonvoid connected] subspaces M′ ⊆ M and N′ ⊆ N with S = M′ × N′.

(iii) The interior Int S_{1} of a subsemigroup S_{1} of S is either void or a subsemigroup of S.

Proof. Assertions (i) and (ii) follow immediately from the definition of the multiplication and the continuity and openness of the projections p_{1}: M×N → M and p_{2}: M × N → N.

(iii) By (ii) we know that S′ = M′ × N′, where M′ ⊆ M, N′ ⊆ N. Since Int(M′ × N′) = Int(M′) × Int(N′) our assertion follows from (ii).

4.14. The diamond product. We now put M = Hor, N = Vert and write Hyp^{−} for Hor × Vert, endowed with the rectangular band structure of 4.12. Then we pull back the multiplication of Hyp^{−} to a partial multiplication, the diamond product ^{♢}, on Hyp. This amounts to:

If hor(X) intersects vert(Y) then the diamond product X^{♢}Y of X and Y is defined as the point of intersection, that is, {X^{♢}Y } = hor(X) ∩ vert(Y).

(i) By the above discussion the diamond product X^{♢}Y is defined for all X, Y ∈ Hyp with hor(X) ≠ − vert(Y), it is jointly continuous and associative as long as all products involved are defined.

(ii) In particular, all vertical and all horizontal lines are semigroups with respect to the diamond product. The horizontal lines are right zero semigroups, and the vertical lines are left zero semigroups with respect to ^{♢}.

(iii) If X^{♢}Y is not defined then Y ∈ − hor(X). If both X^{♢}Y and Y ^{♢}X are not defined then X = −Y .

4.15. Example. The diamond products A^{♢}B and B^{♢}A of A = H + βP and B = H + γQ compute to

A^{♢}B = H, B^{♢}A = \frac{1}{4 + βγ} ((4 − βγ)H + 4βP + 4γQ) .

Note that B^{♢}A exists if and only if βγ ≠ −4. In the special case β = γ = 2 we get A^{♢}B = P + Q (= T in the notation of [5]).

4.16. Remark. The space Hyp^{−} not only bears the natural rectangular band structure of above but also two natural actions of Sl(2, ℝ):

Sl(2, ℝ) × Hyp^{−} → Hyp^{−}, (g, (h, v)) ↦ g · (h, v) = (g · h, v),
Hyp^{−} × Sl(2, ℝ) → Hyp^{−}, ((h, v), g) ↦ (h, v) · g = (h, v · g).

These actions induce partial actions of Sl(2, ℝ) on Hyp, which we also denote with a “·.” For instance, if \binom{a b}{c d} ∈ Sl(2, ℝ) with a ≠ 0 then \binom{a b}{c d}·H = H + \frac{2c}{a}Q, and H · \binom{a b}{c d} = H + \frac{2b}{a}P. We do not use this additional structure as yet, but shall exploit it in [17] for the construction of compactifications.

5. Rectangular domains

5.1. Connected ^{♢} -semigroups.

(i) A ^{♢} -subsemigroup D of Hyp is connected if and only if for every two elements X, Y ∈ D lying on the same horizontal or vertical line the line segment joining X and Y lies in D.

(ii) The intersection of an arbitrary family D_{i} of connected ^{♢} - subsemigroups of Hyp is connected.

Proof. Assertion (i) follows from 4.13(ii), assertion (ii) is then an immediate consequence of (i).

5.2. Rectangular domains. An open connected ^{♢} -semigroup D ⊂ Hyp is called a rectangular domain. Note that in a rectangular domain all diamond products X^{♢}Y are defined. (Thus Hyp itself is not a rectangular domain.) It can be deduced from what is shown below that every rectangular domain is an open domain (in Hyp) in the topological sense, i.e., it is the interior of its closure (Kuratowski, cf.[4], p.37).

5.3. Examples. (i) The intersection D(+) of Hyp with the interior of 𝔰𝔩(2, ℝ)^{+} is a rectangular domain, which is bounded by the four half lines ±H + ℝ_{0}^{+} · P and ±H + ℝ_{0}^{+} · Q. More specifically, a straightforward computation verifies that the map

j: ℝ^{+} × ℝ^{+} ⟶ D(+), (λ, μ) ⟼ \frac{1 − λμ}{1 + λμ} · H + \frac{2μ}{1 + λμ} · P + \frac{2λ}{1 + λμ} · Q

is bijective (with inverse j^{−1}(αH + βP + γQ) = (\frac{1−α}{β}, \frac{β}{1+α})); it is a homomorphism if ℝ^{+} × ℝ^{+} is endowed with the rectangular band structure of 4.12. Note that the identities

\begin{aligned}j(λ, μ) & = H + 2μP + \frac{2λμ}{1 + λμ} · (−H − μP + \frac{1}{μ}Q) \\ & = H + 2λQ + \frac{2λμ}{1 + λμ} · (−H + \frac{1}{λ}P − λQ) \end{aligned}

show that j maps the sets {λ} × ℝ^{+} onto horizontal line segments, the sets ℝ^{+} × {μ} onto vertical line segments.

(ii) If the intersection of an arbitrary family of conjugates of D(+) is nonvoid and open then it is a rectangular domain. This follows from 5.1(ii).

5.4. Basic properties of rectangular domains. For a rectangular domain D the following assertions hold:

(i) Let X ∈ D. Then − hor(X) ∩ D = − vert(X) ∩ D = Ø.

(ii) The closure \overline{c(D)} of c(D) in Hyp^{−} is the product A × B of two proper closed arcs A ⊆ Hor, B ⊆ Vert.

(iii) The nilpotency points of \overline{c(D)}, if they exist, are corner points, i.e., contained in ∂A × ∂B, and no two of them lie on the same bounding arc. Thus \overline{c(D)} contains at most two nilpotency points.

(iv) The interior Int(\overline{D}) of \overline{D} in Hyp coincides with D.

(v) The closure of D is compact if and only if \overline{c(D)} does not contain nilpotency points.

Proof. Assertion (i) follows from the fact that X^{♢}Y is not defined if Y ∈ − hor(X) and Y ^{♢}X is not defined if Y ∈ − vert(X).

(ii) Since D is connected and c is an open and isomorphic embedding we know that c(D) is an open connected subsemigroup of Hyp^{−}. By (i) we can have neither \overline{hor(D)} = Hor nor \overline{vert(D)} = Vert. Thus \overline{c(D)} = A × B, where A is an arc in Hor with two distinct endpoints a_{1} , a_{2} , and B is an arc in Vert with two distinct endpoints b_{1}, b_{2} .

(iii) Since c(D) is the interior of \overline{c(D)}, the nilpotency points of \overline{c(D)} must lie on the boundary of c(D). The boundary of D is the union of the four line segments

h_{i} =^{def} c^{−1}({a_{i}} × B),  v_{j} =^{def} c^{−1}(A × {b_{j}}),  i = 1, 2, j = 1, 2.

Thus every nilpotency point in \overline{c(D)} must be an endpoint of one of the arcs \overline{c(h_{i})}, i = 1, 2. Since \overline{c(h_{i})} contains at most one nilpotency point, this implies the assertion.

(iv) Since c is continuous and open we have Int(\overline{D}) = c^{−1}( Int (\overline{c(D)})), hence the assertion follows from (ii) and (iii).

Assertion (v) is a consequence of 4.9(i).

5.5. Rectangular domains and exponential subsemigroups of Sl(2, ℝ).

(i) If D is a rectangular domain then the set W_{0} = ℝ^{+}D is open and convex, and its closure W = \overline{W_{0}} is a Lie semialgebra. Moreover, exp(W) is a closed subsemigroup of Sl(2, ℝ).

(ii) Conversely, if W is a Lie semialgebra with nonvoid interior in 𝔰𝔩(2, ℝ) and exp W is a semigroup then the intersection D = W_{0} ∩ Hyp of its interior W_{0} = Int W with Hyp is a rectangular domain.

Proof. (i) We first notice that the map

f: ℝ^{+} × Hyp → Kill^{+}, (t, X) ⟼ tX,

is a homeomorphism. Since D is an open subset of Hyp and since f is an open map, the image f(ℝ^{+} × D) = ℝ^{+}D is an open subset of Kill^{+} . Now pick two elements X, Y ∈ D. We write S(X, Y) for the connected ^{♢} -semigroup generated by the line segments conv{X, X^{♢}Y } and conv{X, Y ^{♢}X}. First we will show that ℝ_{0}^{+}S(X, Y) is a Lie semialgebra. If X and Y lie on the same horizontal or vertical line then ℝ_{0}^{+}S(X, Y) is a wedge in the Borel algebra generated by S(X, Y), hence it is a Lie semialgebra. Suppose now that X and Y do not lie on the same horizontal or vertical line. Then the polyhedral wedge

C = ℝ_{0}^{+}X + ℝ_{0}^{+}X^{♢}Y + ℝ_{0}^{+}Y ^{♢}X + ℝ_{0}^{+}Y

is the intersection of four closed half spaces, each bounded by one of the four Borel algebras generated by the two dimensional faces

ℝ_{0}^{+}X + ℝ_{0}^{+}X^{♢}Y, ℝ_{0}^{+}X + ℝ_{0}^{+}Y ^{♢}X, ℝ_{0}^{+}Y + ℝ_{0}^{+}X^{♢}Y, ℝ_{0}^{+}Y + ℝ_{0}^{+}Y ^{♢}X.

Thus C is a Lie semialgebra. Moreover, since the boundary of C does not contain nonzero nilpotent elements, we conclude that C \ {0} ⊆ Kill^{+} .

Since f is a homeomorphism, it maps the boundary ∂(ℝ^{+} × S(X, Y)) = ℝ^{+} × ∂(S(X, Y)) of ℝ^{+} × S(X, Y) in ℝ^{+} × Hyp onto the boundary of f(ℝ^{+} × S(X, Y)) in Kill^{+} . Hence

ℝ^{+}∂(S(X, Y)) = ∂(ℝ^{+}S(X, Y)).

On the other hand, the boundary of C \ {0} in Kill^{+} is exactly ℝ^{+}∂(S(X, Y)). Thus C \ {0} and ℝ^{+}S(X, Y) have the same boundary in Kill^{+} . Also, these two sets have non-empty intersection. Since both of these sets have dense and connected interior, they must coincide, i.e., ℝ^{+}S(X, Y) = C \ {0}. Hence ℝ_{0}^{+}S(X, Y) = C, which shows that ℝ_{0}^{+}S(X, Y) is a Lie semialgebra.

In particular, we see that W_{0} = ℝ^{+}D is convex, so W = \overline{W_{0}} is a wedge. Since every two elements of W_{0} lie in a Lie semialgebra contained in W, Corollary II.2.16 of [5], p. 89, applies and shows that W is a Lie semialgebra. Also, W ⊆ \overline{Kill^{+}} , hence exp(W) is a closed subsemigroup of Sl(2, ℝ), by 3.8.

(ii) Suppose that W is a wedge with nonvoid interior in 𝔰𝔩(2, ℝ) such that exp W is a semigroup. Then by 3.8 W is the intersection of certain conjugates of 𝔰𝔩(2, ℝ)^{+} . Note that for any two elements X, Y ∈ 𝔰𝔩(2, ℝ)^{+} ∩ Hyp the diamond product X^{♢}Y , if defined, also lies in 𝔰𝔩(2, ℝ)^{+} . It follows that W ∩Hyp is closed under the ^{♢} -product, as far as the latter is defined. The diamond product is defined for all X, Y in the interior W_{0} ∩ Hyp = Int(W ∩ Hyp) (since W_{0} ∩ Hyp lies in a conjugate of Int(𝔰𝔩(2, ℝ)^{+}) ∩ Hyp). Recalling that the diamond product is an open map (4.13(i)) we see that W_{0} ∩ Hyp is a ^{♢} -semigroup. The inverse image f^{−1}(W_{0}) = ℝ^{+} × (W_{0} ∩ Hyp) of W_{0} under the map f in the proof of (i) is connected, so W_{0} ∩ Hyp is connected. Thus W_{0} ∩ Hyp is a rectangular domain.

5.6. Remark. Since the boundary of a rectangular domain consists of two horizontal and two vertical line segments the above assertion (ii) implies that the Lie wedge of an exponential subsemigroup of Sl(2, ℝ) with inner points is the intersection of at most four half spaces bounded by Borel algebras. This assertion was formulated first in [5] (p.110; its proof was left to the reader as exercise E.II.1).

5.7. Nilpotency points and nilpotent elements. Let D be a rectangular domain and write W_{0} = ℝ^{+}D, W = \overline{W_{0}} .

(i) If \overline{c(D)} contains a nilpotency point (a, −a) then the Borel algebra 𝔟 = \overline{ℝ · a} intersects W in a closed face which in its algebraic interior contains a nonzero nilpotent element N.

(ii) Conversely, if W contains a nonzero nilpotent element N then N lies in the algebraic interior of a closed two dimensional face of W and \overline{c(D)} contains a nilpotency point (a, −a), where a is the horizontal line contained in the Borel algebra generated by N.

(iii) Let k be the cardinality of the set of nilpotency points in \overline{c(D)}. Then W_{0} = ℝ^{+}D is the intersection of exactly 4 − k distinct open half spaces in 𝔰𝔩(2, ℝ), each of which is bounded by a Borel algebra. The closed wedge W contains exactly k linearly independent nilpotent elements.

Proof. (i) Let X ∈ a∩W, Y ∈ −a∩W. Then X, Y are nonzero and lie in the boundary of W, also N = X + Y is a nilpotent element (by 4.4(iii)). If N ≠ 0 then the assertion follows. If N = 0 then X = −Y ∈ H(W) and thus W ∩ 𝔟 is one of the two closed half spaces of 𝔟 bounded by ℝX, and these must contain a nonzero nilpotent element.

(ii) Suppose that N is a nonzero nilpotent element in W. Then N ∈ ∂W, since Int W ⊂ Kill^{+} . Let F be a closed two dimensional face of W with N ∈ F. We have seen in 5.6 that F = 𝔟 ∩ W, where 𝔟 is a Borel algebra. Since the Borel algebra 𝔟 passing through N is uniquely defined, we conclude that N must be contained in the algebraic interior of F. It follows that \overline{D} meets both the horizontal line a and the vertical line −a in 𝔟, and thus \overline{c(D)} must contain the nilpotency point (a, −a).

The proof of (iii) is left to the reader.

6. Examples of connected subsemigroups of Sl(2, ℝ)

6.1. In this section we discuss a collection of typical examples of connected subsemigroups of Sl(2, ℝ). Some of these will be used later to illustrate the application of our main results.

The main focus of attention is given to three dimensional exponential subsemigroups and their Lie wedges. In fact, we shall see in the next section that our examples include a representative of each conjugacy class of three dimensional exponential subsemigroups.

It is instructive (and very convenient, too) to represent our examples as compression semigroups, some aspects of this approach, in particular the observation that all semigroups of our list are defined on ‘semialgebraic’ sets, seem to be of independent interest.

6.2. Compression semigroups. Let S be a semigroup which acts on some space X. Then for every subset M of X we define the compression semigroup in S of M as the set

compr_{S}(M) =^{def} {s ∈ S | sM ⊆ M}.

The original definition of compression semigroup (as used, e.g., in [6], p.203) supposes that S is a group, since the main application is to find maximal subsemigroups of a group. In our present setting S will usually be a subsemigroup of G = Sl(2, ℝ) (acting on ℝ^{2} ) and it will make a difference whether the compression semigroup is formed only within S or in the whole group G. In general, an inclusion S ⊆ S′ implies a corresponding inclusion compr_{S}(M) ⊆ compr_{S′}(M), but this inclusion is proper in most cases, even if S has inner points in S′ (cf. the example in 6.10 below).

6.3. Remark. The following observation is checked easily by straightforward calculation.

Let S be a subsemigroup of a group G which acts on a space X. Then for any subset Y ⊆ X and a ∈ G we have

(i) compr_{G}(aY) = a compr_{G}(Y)a^{−1} ;

(ii) compr_{S}(aY) = S ∩ a compr_{G}(Y)a^{−1} .

6.4. Example. The semigroup Sl(2, ℝ)^{+} , with Lie wedge 𝔰𝔩(2, ℝ)^{+} , is the compression semigroup in Sl(2, ℝ) of the cone ℝ_{0}^{+} × ℝ_{0}^{+} . Note that Sl(2, ℝ)^{+} = exp (𝔰𝔩(2, ℝ)^{+}) is an exponential semigroup.

6.5. Example. It is well known (cf.[5], p.419) that the set

Sl(2, ℝ)^{++} =^{def} \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Sl(2, ℝ)^{+} | 1 ≤ a\}

is a closed subsemigroup of Sl(2, ℝ)^{+} with Lie wedge W = ℝ_{0}^{+}H + ℝ_{0}^{+}P + ℝ_{0}^{+}Q. Also, the decomposition

( \begin{matrix}a & b \\ c & d \end{matrix} ) = ( \begin{matrix}1 & 0 \\ a^{−1}c & 1 \end{matrix} ) ( \begin{matrix}a & b \\ 0 & a^{−1} \end{matrix} )

shows that Sl(2, ℝ)^{++} is the Lie semigroup generated by W. It is checked easily that Sl(2, ℝ)^{++} is the compression semigroup in Sl(2, ℝ) of the set M_{x} =^{def} {(x, y) ∈ ℝ^{2} | x ≥ 1, y ≥ 0}.

Likewise the set

Sl(2, ℝ)^{+−} =^{def} \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Sl(2, ℝ)^{+} | 1 ≤ d\}

is a Lie subsemigroup of Sl(2, ℝ)^{+} with Lie wedge W = −ℝ_{0}^{+}H + ℝ_{0}^{+}P + ℝ_{0}^{+}Q, and this semigroup is the compression semigroup in Sl(2, ℝ) of the set M_{y} =^{def} {(x, y) ∈ ℝ^{2} | x ≥ 0, y ≥ 1}.

The semigroup Sl(2, ℝ)^{+−} is the image of Sl(2, ℝ)^{++} under the (outer) automorphism which, on the level of Lie algebras, takes H to −H and interchanges P and Q. Both of these semigroups fail to be exponential.

6.6. Example. Now we form the intersection of the two semigroups in the preceding example:

Sl(2, ℝ)^{++,+−} =^{def} \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Sl(2, ℝ)^{+} | 1 ≤ a and 1 ≤ d\}

= \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Sl(2, ℝ)^{+} | 1 ≤ a ≤ 1 + bc\}

is a closed subsemigroup of Sl(2, ℝ)^{+} with Lie wedge W = ℝ_{0}^{+}P + ℝ_{0}^{+}Q. We claim that every element s = \binom{a b}{c d} ∈ Sl(2, ℝ)^{++,+−} decomposes in the form

s = exp(qQ) exp(pP) exp(q′Q) ∈ exp(W)^{3}, with p, q, q′ ∈ ℝ_{0}^{+},

so Sl(2, ℝ)^{++,+−} = exp(W)^{3} . Indeed, this decomposition is trivial if b = 0 (and thus a = d = 1), whereas for b > 0 we have the decomposition

( \begin{matrix}a & b \\ c & d \end{matrix} ) = ( \begin{matrix}1 & 0 \\ \frac{d−1}{b} & 1 \end{matrix} ) ( \begin{matrix}1 & b \\ 0 & 1 \end{matrix} ) ( \begin{matrix}1 & 0 \\ \frac{a−1}{b} & 1 \end{matrix} ) .

6.7. Example. The compression semigroup compr_{G}(M_{x} ∩ M_{y}) of the intersection M_{x} ∩ M_{y} = {(x, y) ∈ ℝ^{2} | x ≥ 1, y ≥ 1} in G = Sl(2, ℝ) is the semigroup

\{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Sl(2, ℝ)^{+} | a + b ≥ 1 and c + d ≥ 1\} ,

which properly contains Sl(2, ℝ)^{++,+−}. (For example, the matrix \binom{\frac{1}{2} 1}{1 4} belongs to this compression semigroup, but not to Sl(2, ℝ)^{++,+−}.)

6.8. Example. For a fixed real λ > 0 we define the cone

C_{λ} = {(x, y) ∈ ℝ^{2} | x ≥ λy ≥ 0}

and write S_{λ} for the compression semigroup of C_{λ} in Sl(2, ℝ)^{+} . Then

(∗)  \begin{aligned}S_{λ} & = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Sl(2, ℝ)^{+} a + \frac{1}{λ}b ≥ λc + d\} \\ & = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Sl(2, ℝ)^{+} (λa + b)(a − λc) ≥ λ\} . \end{aligned}

Indeed, for \binom{a b}{c d} ∈ Sl(2, ℝ)^{+} the inequalities

a + b/λ ≥ λc + d and (λa + b)(a − λc) ≥ λ

are equivalent, since multiplying the first inequality with a > 0 yields (note that ad = 1 + bc)

a(a + b/λ) ≥ λac + ad = λac + 1 + bc = 1 + λc(a + b/λ),

and this is equivalent to (λa + b)(a − λc) ≥ λ. Thus in (∗) the two sets on the right are the same.

Looking at the special case x = 1, y = 1/λ we see that for every matrix s = \binom{a b}{c d} ∈ Sl(2, ℝ)^{+} with sC_{λ} ⊆ C_{λ} the inequality a + b/λ ≥ λc + d holds. Conversely, if s = \binom{a b}{c d} ∈ Sl(2, ℝ)^{+} with (λa + b)(a − λc) ≥ λ then a − λc > 0 and

s ( \begin{matrix}1 \\ 1/λ \end{matrix} ) = ( \begin{matrix}a + b/λ \\ c + d/λ \end{matrix} ) ∈ C_{λ} as well as s ( \begin{matrix}1 \\ 0 \end{matrix} ) = ( \begin{matrix}a \\ c \end{matrix} ) ∈ C_{λ},

therefore sC_{λ} = s(ℝ_{0}^{+} ( 1_{\frac{1}{λ}} ) + ℝ_{0}^{+} ( \begin{matrix}1 \\ 0 \end{matrix} )) ⊆ C_{λ}.

Note that all semigroups S_{λ} are conjugate to S_{1} :

S_{λ} = ( \begin{matrix}\sqrt{λ} & 0 \\ 0 & 1/\sqrt{λ} \end{matrix} ) S_{1} ( \begin{matrix}1/\sqrt{λ} & 0 \\ 0 & \sqrt{λ} \end{matrix} ) ,

and that S_{λ} ⊆ S_{μ} whenever λ ≥ μ > 0.

We now compute the Lie wedge of S_{1} using the following one parameter subsemigroups of S_{1} :

S_{A} = \{( \begin{matrix}a & 0 \\ a − 1/a & 1/a \end{matrix} ) | a ≥ 1\} with infinitesimal generator A = 2Q + H,
S_{H} = \{( \begin{matrix}h & 0 \\ 0 & 1/h \end{matrix} ) | h ≥ 1\} with infinitesimal generator H,
S_{B} = \{( \begin{matrix}1/b & b − 1/b \\ 0 & b \end{matrix} ) | b ≥ 1\} with infinitesimal generator B = 2P − H.

For every element s = \binom{u v}{w x} ∈ S_{1} we have the decomposition

s = ( \begin{matrix}a & 0 \\ a − 1/a & 1/a \end{matrix} ) ( \begin{matrix}h & 0 \\ 0 & 1/h \end{matrix} ) ( \begin{matrix}1/b & b − 1/b \\ 0 & b \end{matrix} ) ,

where

a = \sqrt{\frac{u}{u − w}} , b = \sqrt{\frac{u + v}{u}} , h = \sqrt{(u + v)(u − w)}

(recall that by (∗) the inequality u+v ≥ x+w is equivalent to (u+v)(u−w) ≥ 1). Thus the Lie wedge W_{1} of S_{1} is spanned by the three vectors A,B,H:

W_{1} = ℝ_{0}^{+}H + ℝ_{0}^{+}(H + 2Q) + ℝ_{0}^{+}(−H + 2P),

and S_{1} is the Lie semigroup generated by W_{1} ,

S_{1} = S_{A}S_{H}S_{B} = S_{H}S_{A}S_{B} = S_{H}S_{B}S_{A}.

Note also that S_{A}S_{B} = S_{B}S_{A}, S_{A}S_{H} = S_{H}S_{A}, but S_{H}S_{B} ⫌ S_{B}S_{H}. Since W_{1} is a Lie semialgebra contained in \overline{Kill^{+}} we deduce from 3.8 that S_{1} is exponential.

6.9. Remark. It is not difficult to deduce right from its definition that S_{1} is divisible:

Since the dyadic fractions are dense in ℝ, and since the restriction of exp to \overline{Kill^{+}} is a diffeomorphic embedding, it suffices to show that if for some s = \binom{a b}{c d} ∈ Sl(2, ℝ)^{+} the square s^{2} ∈ S_{1} then s ∈ S_{1} . Applying the definition of S_{1} to

s^{2} = ( \begin{matrix}a^{2} + bc & b(a + d) \\ c(a + d) & d^{2} + bc \end{matrix} )

we see that s^{2} ∈ S_{1} implies a^{2} + bc + b(a + d) ≥ c(a + d) + d^{2} + bc, which is equivalent to (a + d)(a − d + b − c) ≥ 0, or, since a + d > 0, to a + b ≥ c + d.

6.10. Example. The compression semigroup of C_{1} with respect to the whole group G = Sl(2, ℝ) properly contains S_{1} . Since

C_{1} = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) (ℝ_{0}^{+} × ℝ_{0}^{+})

Example 6.4 and Remark 6.3 show that compr_{G}(C_{1}) is conjugate to Sl(2, ℝ)^{+} , namely

compr_{G}(C_{1}) = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) Sl(2, ℝ)^{+} ( \begin{matrix}1 & −1 \\ 0 & 1 \end{matrix} ) .

This set contains, e.g., the element \binom{1 1}{0 1}\binom{2 1}{1 1}\binom{1 −1}{0 1} = \binom{3 −1}{1 0} ∉ S_{1} . Furthermore,

S_{1} = Sl(2, ℝ)^{+} ∩ ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) Sl(2, ℝ)^{+} ( \begin{matrix}1 & −1 \\ 0 & 1 \end{matrix} ) .

A more specific (but straightforward) calculation shows that

compr_{G}(C_{1}) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Sl(2, ℝ) | a + b ≥ c + d ≥ 0 and a ≥ c ≥ 0\} .

6.11. Example. The anti-isomorphism sending every matrix s to its transpose s^{T} maps S_{1} onto the semigroup

\begin{aligned}S^{1} & = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Sl(2, ℝ)^{+} | a + c ≥ b + d\} \\ & = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Sl(2, ℝ)^{+} | (a − b)(a + c) ≥ 1\} . \end{aligned}

Its Lie wedge is L(S^{1}) = ℝ_{0}^{+} · H + ℝ_{0}^{+} · A + ℝ_{0}^{+} · B, where A = 2P + H, B = 2Q − H. This semigroup is the compression semigroup of the dual cone C^{1} = {(x, y) | x ≥ 0, y ≥ −x} of C_{1} .

More generally, the map s ↦ s^{T} maps each semigroup S_{λ} onto a semigroup S^{λ}. Each S^{λ} is the compression semigroup in Sl(2, ℝ)^{+} of the cone C^{λ} = {(x, y) | x ≥ 0, y ≥ −λx} and all semigroups S^{λ} are conjugate to S^{1} .

6.12. Example. For λ, μ ∈ ℝ^{+} we let S_{λ}^{μ} = S_{λ} ∩ S^{μ}. Then S_{λ}^{μ} is the compression semigroup in Sl(2, ℝ)^{+} of the union C_{λ} ∪ C^{μ} = {(x, y) ∈ ℝ^{2} | x ≥ 0, −μx ≤ y ≤ \frac{1}{λ}x\}. Note that every semigroup S_{λ}^{μ} is exponential. Also, S_{λ}^{μ} is conjugate to S_{λ′}^{μ′} if and only if λμ = λ′μ′. The Lie wedge of S_{λ}^{1} is spanned by the vectors H, H + 2P, H + \frac{2}{λ}Q and (λ − 1)H + 2λP + 2Q (the Lie wedges for general λ, μ can be deduced readily). We leave the straightforward proofs to the reader.

6.13. Semialgebraic Semigroups. Recall that a subset of ℝ^{n} is called a semialgebraic set if it is the solution set (in some ℝ^{n}) of finitely many polynomial equalities and inequalities, or a finite union of such sets (cf. [2], p.162). It is natural to say that a semigroup S is a semialgebraic semigroup, if it is defined on a semialgebraic set and the multiplication is defined by polynomial functions. In this sense all semigroups of this section are semialgebraic. More generally, if we have an algebraic action of a semialgebraic semigroup S then every compression semigroup in S of a semialgebraic set is a semialgebraic semigroup.

7. Conjugacy classes of exponential Lie wedges in 𝔰𝔩(2, ℝ)

7.1. In this section we give a complete list of the conjugacy classes of Lie semialgebras with nonvoid interior in \overline{Kill^{+}} . Since by 5.5 these Lie semialgebras are in 1-1-correspondence with the rectangular domains in Hyp this amounts to giving a list of the conjugacy classes of rectangular domains in Hyp. A first natural classification of the rectangular domains in Hyp is based on counting the number of nilpotency points.

7.2. The three types of rectangular domains. A rectangular domain D is said to be of infinity type k, or a type k rectangular domain for short, if \overline{c(D)} contains exactly k nilpotency points in Hyp^{−}. (Note that 0 ≤ k ≤ 2.)

We say that a wedge W is a semialgebra of infinity type k, or a type k semialgebra for short, if Int W = ℝ^{+}D, where D is a type k rectangular domain.

Proposition 5.7 says that every rectangular domain belongs to exactly one of the above types. Also, a Lie semialgebra W is of type k if and only if it does not meet the interior of the light cone Kill^{0} and is the intersection of exactly 4−k half space semialgebras (or equivalently: contains exactly k linearly independent nilpotent elements).

7.3. The rectangular domains of infinity type 2. Every type 2 rectangular domain is conjugate to Int(𝔰𝔩(2, ℝ)^{+}) ∩ Hyp. Proof. This is an immediate consequence of the well known fact that all Lie semialgebras (indeed all Lie wedges) in 𝔰𝔩(2, ℝ) with nontrivial edge and nonvoid interior are conjugate to 𝔰𝔩(2, ℝ)^{+} (cf., e.g., [5], p.109f).

7.4. The rectangular domains of infinity type 1. Every type 1 rectangular domain is conjugate to the rectangular domain

(i) D_{1} with the three corner points H, H + 2P, H − 2Q or to

(ii) the transpose D_{1}^{T} of D_{1} , with the three corner points H, H − 2P, H + 2Q.

Proof. Let D be a rectangular domain of type 1. Then by assumption, the boundary of D in Hyp has exactly three corner points, say A, B, C. We may assume that A = B^{♢}C. Upon applying a suitable inner automorphism we enforce that A = H, so that B = H + λP and C = H + μQ for some nonzero λ, μ ∈ ℝ. Also, the diamond product C^{♢}B does not exist, hence, as we have seen in Example 4.15, we must have λμ = −4. Now the inner automorphism e^{tadH} with t = 1/2(log 2 − log |λ|) carries the triple (A, B, C) either to the triple (H, H + 2P, H − 2Q) or to the triple (H, H − 2P, H + 2Q). It follows that D is conjugate either to D_{1} or to D_{1}^{T} .

(Note that D_{1}^{T} is mapped to −D_{1} by conjugation with the rotation matrix g = \binom{0−1}{1 0}.)

7.5. For the list of conjugacy classes of type 0 rectangular domains we need a device which to every such domain assigns a convenient reference point and an area.

(i) Consider the element H ∈ Hyp. We noticed already in 4.6 that the inner automorphisms of 𝔰𝔩(2, ℝ) act transitively on the space Hyp, so for every X ∈ Hyp there is an inner automorphism φ carrying X to H. For every such inner automorphism φ we also have φ(hor(X)) = hor(H) = H + ℝP and φ(vert(X)) = vert(H) = H + ℝQ.

(ii) The inner automorphisms fixing H are exactly the maps e^{tadH}, t ∈ ℝ, and these act transitively on each connected component of hor(H) \ {H}. Thus if X, Y ∈ Hyp with Y ∈ hor(X) \ {X} then there is exactly one inner automorphism φ with φ(X) = H and either φ(Y) = H + P or φ(Y) = H − P.

(iii) Similarly, if X, Y ∈ Hyp with Y ∈ vert(X) \ {X} then there is a unique inner automorphism ψ such that ψ(X) = H and either ψ(Y) = H + Q or ψ(Y) = H − Q.

7.6. Orientation. For any X ∈ Hyp we say that

(i) an element Y ∈ Hyp lies on the right of X, or X lies on the left of Y , in symbols: X <_{h} Y , if there exists an inner automorphism of 𝔰𝔩(2, ℝ) sending X to H and Y to H + P;

(ii) an element Y ∈ Hyp lies above X, or X lies below Y , in symbols: X <_{v} Y , if there exists an inner automorphism of 𝔰𝔩(2, ℝ) sending X to H and Y to H + Q.

7.7. Remark. (i) On each horizontal line the relation <_{h} induces a total order. (Note that there is an inner automorphism α with α(H − P) = H and α(H) = H + P.) Similarly, <_{v} induces a total order on every vertical line.

(ii) The partial orders induced by <_{h} and <_{v} are preserved under every inner automorphism. (The outer automorphisms reverse them.)

7.8. Lower left corner and characteristic of a type 0 rectangular domain. Let D be a type 0 rectangular domain. Then the boundary of D in Hyp has four corners. The corner point A of ∂D will be called the lower left corner of D if A <_{h} B for every B ∈ ∂D ∩ hor(A) and A <_{v} C for every C ∈ ∂D∩vert(A). It is immediate that a lower left corner of D exists and that it is uniquely defined. Since by definition inner automorphisms respect the ordering on horizontal or vertical lines, the lower left corner of a rectangular domain D is carried to the lower left corner of φ(D) by every inner automorphism φ of 𝔰𝔩(2, ℝ).

We now assign to each type 0 rectangular domain D a characteristic δ(D). Let A be the lower left corner of D and let B, C be the corner points adjacent to A, so that B ∈ hor(A), C ∈ vert(A). Then the Lie bracket [B − A, C − A] is a positive multiple r · A of A, we define δ(D) = \sqrt{r}. Note that δ is invariant under inner automorphisms. For A = H the number δ(D)^{2} is just the Euclidean area of the rectangle spanned by the vectors B − A and C − A.

7.9. Remark. The characteristic δ(D) can be computed also in terms of opposite corners of D (without knowing the lower left corner of D), after conjugating one of them onto H. If H and Y = αH + βP + γQ are opposite corners of D then δ(D) = \frac{2}{1+α} \sqrt{|βγ|}. (We shall see in 8.8 that 1 + α > 0.) Observing that H^{♢}Y = H + \frac{2β}{1+α}P and Y ^{♢}H = H + \frac{2γ}{1+α}Q this formula is checked by straightforward calculation.

7.10. The conjugacy classes of type 0 rectangular domains. Let D be a rectangular domain of type 0 with δ(D) = λ. Then D is conjugate to the rectangular domain D_{0}(λ) with the four corner points

A = H, B = H + λP, C = H + λQ,

C^{♢}B = \frac{1}{4 + λ^{2}} ((4 − λ^{2})H + 4λP + 4λQ) .

Proof. Applying a suitable inner automorphism we enforce that H is the lower left corner of D. Then, similar to the proof of 7.4, we apply an inner automorphism e^{tadH} for a suitable real t which maps D onto D_{0}(λ).

7.11. Conjugacy classes of exponential Lie semialgebras. Let W be a Lie semialgebra with nonvoid interior which is contained in \overline{Kill^{+}} . Then exactly one of the following assertions holds:

(i) W is conjugate to 𝔰𝔩(2, ℝ)^{+} ;

(ii) W is conjugate either to

W (D_{1}) = ℝ_{0}^{+}H + ℝ_{0}^{+}(H + 2P) + ℝ_{0}^{+}(H − 2Q)

or to

W (D_{1}^{T}) = ℝ_{0}^{+}H + ℝ_{0}^{+}(H − 2P) + ℝ_{0}^{+}(H + 2Q);

(iii) there exists a unique positive real number λ such that W is conjugate to W (D_{0}(λ)) = ℝ_{0}^{+}A + ℝ_{0}^{+}B + ℝ_{0}^{+}C + ℝ_{0}^{+}C^{♢}B, with

A = H, B = H + λP, C = H + λQ,

C^{♢}B = \frac{1}{4 + λ^{2}} ((4 − λ^{2})H + 4λP + 4λQ) .

Proof. This is immediate from the above classification of rectangular domains in view of the 1-1 correspondence between rectangular domains and Lie semialgebras with nonvoid interior in \overline{Kill^{+}} .

7.12. Lie wedges of exponential compression semigroups. In section 4 we have seen that the compression semigroups S_{λ}, S^{μ} and S_{λ}^{μ} = S_{λ} ∩ S^{μ} are exponential. We now explicitly indicate the conjugacy classes of the corresponding Lie wedges. The following assertions can be checked by straightforward calculation:

(i) The Lie wedge W_{λ} of S_{λ} is conjugate to W (D_{1}). In fact,

W_{λ} = u_{λ}W (D_{1})u_{λ}^{−1} with u_{λ} = ( \begin{matrix}\sqrt{λ} & 0 \\ 0 & \frac{1}{\sqrt{λ}} \end{matrix} ) ( \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} ) .

(ii) The Lie wedge W^{μ} of S^{μ} is conjugate to W (D_{1}^{T}), with conjugating element (u_{μ}^{−1})^{T} .

(iii) The Lie wedge W_{λ}^{μ} of S_{λ}^{μ} is conjugate to W (D_{0}(\frac{2}{\sqrt{λμ}})). In fact,

W_{λ}^{μ} = vW (D_{0}(\frac{2}{\sqrt{λμ}}))v^{−1} with v = ( \begin{matrix}\sqrt[4]{\frac{λ}{μ}} & 0 \\ 0 & \frac{1}{\sqrt[4]{\frac{μ}{λ}}} \end{matrix} ) .

7.13. The above list shows that every conjugacy class of exponential Lie semialgebras with nonvoid interior in 𝔰𝔩(2, ℝ) contains a representative which is the Lie wedge of an exponential compression semigroup. The following proposition formulates this fact in terms of exponential semigroups.

7.14. Conjugacy classes of exponential subsemigroups. Let S be a three dimensional exponential subsemigroup of Sl(2, ℝ). Then S is conjugate to exactly one of the following semigroups:

(i) Sl(2, ℝ)^{+} ;

(ii) S_{1} = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Sl(2, ℝ)^{+} | a + b ≥ c + d\};

(iii) S^{1} = (S_{1})^{T} = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Sl(2, ℝ)^{+} | a + c ≥ b + d\};

(iv) S_{λ}^{1} = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Sl(2, ℝ)^{+} | a + c ≥ b + d and a + \frac{1}{λ}b ≥ λc + d\}, for

some λ ∈ ℝ^{+} .

7.15. Remark. The above proposition says that all three dimensional exponential subsemigroups of Sl(2, ℝ) are semialgebraic semigroups in the sense of 6.13. (The exponential subsemigroups of dimension 0,1,2 are trivially semialgebraic.)

8. An asymptotic formula for the diamond product

8.1. Proposition. For two points X, Y in Hyp the following assertions are equivalent:

(i) There exist a positive real s and a bound B ∈ ℕ such that for all natural numbers m > B and n > B the product exp(msX) exp(nsY) lies in exp(Kill^{+}).

(ii) There exists a positive real s such that exp(sX) and exp(sY) (algebraically) generate a semigroup which is contained in exp(Kill^{+}).

(iii) The semigroup generated by exp(ℝ^{+}X) ∪ exp(ℝ^{+}Y) is contained in exp(Kill^{+}).

(iv) There exists a Lie semialgebra W ⊆ \overline{Kill^{+}} containing both X and Y in its interior.

(v) X and Y lie in the same connected component of the open set Hyp \τ, where τ denotes the tangent plane of Hyp at −X. (Note that τ ∩ Hyp = hor(−X) ∪ vert(−X).)

(vi) The set ℝ^{+}X + ℝ^{+}Y (equivalently, the line segment conv{X, Y }) does not meet the light cone Kill^{0} .

Proof. The implications (iii) ⟹ (ii) ⟹ (i) are obvious. To see the implication (iv) ⟹ (iii) we only have to recall that exp W is a closed semigroup if W is a Lie semialgebra with nonvoid interior contained in \overline{exp(Kill^{+})}, and that the restriction of exp to \overline{Kill^{+}} is a homeomorphic embedding.

In order to establish the implications (i) ⟹ (v), (v) ⟹ (iv), and (v)⟺(vi) we simplify matters by applying a suitable inner automorphism so that X = H. We write Y = αH + βP + γQ. Then the tangent plane of Hyp at −X is the plane −X + ℝ · P + ℝ · Q, so (v) is equivalent to α > −1.

Pick λ, μ ∈ ℝ and let \binom{a b}{c d} = exp(μY). Using the formulas for exp we compute the number τ(λ, μ) = trace(exp λX exp μY): (∗)

τ(λ, μ) = ae^{λ} + de^{−λ} = (cosh(μ) + α sinh(μ))e^{λ} + (cosh(μ) − α sinh(μ))e^{−λ}
= (1 + α) cosh(λ + μ) + (1 − α) cosh(λ − μ).

(i) ⟹ (v) By 2.3 we know that a matrix lies in exp(Kill^{+}) if and only if its trace is > 2, so assertion (i) implies that τ(λ, μ) > 2 for any λ = ms, μ = ns with m, n ∈ B + ℕ. But if α < −1 then lim_{λ→∞} τ(λ, λ) = −∞, and if α = −1 then τ(λ, λ) = 2. Thus we see that α > −1, and therefore (v) holds.

(v) ⟹ (iv) Case 1: |α| < 1. Then we conclude from α^{2}+βγ = 1 that β and γ have the same sign. Applying an inner automorphism of the form e^{h ad H}, h ∈ ℝ, we enforce that β = γ. Then both X = H and Y = αH + β(P + Q) lie in the plane ℝH + ℝ(P + Q). Thus a suitable rotation e^{σ ad(P −Q)} will map 𝔰𝔩(2, ℝ)^{+} onto a semialgebra W which contains both X and Y in its interior.

Case 2: α = 1. Then either β = 0 or γ = 0, so Y = H + βP or Y = H +γQ. Then a rotation e^{σ ad(P −Q)} will map 𝔰𝔩(2, ℝ)^{+} onto a semialgebra which contains both X and Y . (For instance, if Y = H + 2P then σ = \frac{π}{8} will do the job. This situation can be reached by applying suitable automorphisms and/or anti-automorphisms.)

Case 3: α > 1. Then β and γ have opposite sign. Using a suitable inner automorphism e^{h ad H} we enforce that β = −γ. Then α^{2} +βγ = 1 implies that |β| < α. A straightforward calculation shows that both X = H and Y = αH + β(P − Q) lie in the interior of the semialgebra ℝ(P + Q) + ℝ_{0}^{+}(H + P − Q) + ℝ_{0}^{+}(H − P + Q). In fact, H = \frac{1}{2}(H + P − Q) + \frac{1}{2}(H − P + Q) and αH + β(P − Q) = \frac{α+β}{2} (H + P − Q) + \frac{α−β}{2} (H − P + Q).

(v)⟺(vi) Let s and t be positive reals. Then sX + tY = (s + αt)H + βtP + γtQ ∈ Kill^{0} if and only if (s − t)^{2} + 2st(α + 1) = 0, i.e., α + 1 = −\frac{(s−t)^{2}}{2st} . This implies the equivalence of assertions (v) and (vi).

8.2. Remarks. For later use we record the following observation of the above proof: If X = H and Y = αH + βP + γQ then assertion 8.1(v) just means that α > −1.

8.3. Scholium. Suppose that X, Y ∈ Hyp do not belong to the same Borel algebra. Then the assertions listed in 8.1 are also equivalent to

(i′) There exist positive reals s, t and a bound B ∈ ℕ such that for all natural numbers m > B and n > B the product exp(msX) exp(ntY) lies in exp(Kill^{+}).

Proof. The implication (i) ⟹ (i′) is obvious. To see that (i′) ⟹ (v), we suppose, without losing generality, that X = H and Y = αH + βP + γQ. Then we choose two sequences ⟨m_{k}⟩ and ⟨n_{k}⟩ of natural numbers such that lim m_{k} = lim n_{k} = ∞ and lim(m_{k}s − n_{k}t) = 0. Let λ_{k} = m_{k}s and μ_{k} = n_{k}t. If α < −1 then relation (∗) of the proof of 8.1 implies that lim τ(λ_{k}, μ_{k}) = −∞. Since (i′) holds and since X and Y do not belong to the same Borel algebra, we conclude that α > −1.

8.4. Scholium. For two points X, Y ∈ Hyp the following assertions are equivalent:

(i) The line segment conv{X, Y } meets the interior Kill^{−} of the light cone. (ii) For any choice of positive reals s, t there exist natural numbers m, n ∈ ℕ

such that the product exp(msX) exp(ntY) does not belong to the image of the exponential function.

(iii) There exist positive reals s, t and natural numbers m, n ∈ ℕ such that the product exp(msX) exp(ntY) does not belong to the image of the exponential function.

Proof. The implication (ii) ⟹ (iii) is obvious.

(i) ⟹ (ii) Assertion (i) implies that X and Y cannot lie in the same Borel algebra. Choose m_{k} and n_{k} as in the proof of Scholium 8.3. Then for large k the trace τ(m_{k}s, n_{k}t) of exp(m_{k}sX) exp(n_{k}tY) is < −2, so assertion (ii) follows from 2.3.

(iii) ⟹ (i) If (iii) holds then the line segment conv{X, Y } cannot be tangent to the light cone, otherwise X and Y would lie in the same Borel algebra 𝔟, which is impossible since exp 𝔟 is a group. Thus the implication (iii) ⟹ (i) follows from 8.1.

8.5. Proposition. Let X, Y be two elements in Hyp, not belonging to the same Borel algebra. Then the following assertions hold:

(i) There exists a bound B > 0 such that exp(sX) exp(tY) ∉ exp(Kill^{0}), for all s, t > B.

(ii) Let s, t ∈ ℝ^{+} such that the elements exp(sX) and exp(tY) generate a semigroup S with S ⊂ \overline{exp(Kill^{+})}. Then X and Y satisfy the equivalent conditions of 8.1.

Proof. (i) Pick s, t ∈ ℝ^{+} , and, for notational convenience, put exp(tY) = \binom{a b}{c d}. By 2.3(v) we have to show that the trace of exp(sX) exp(tY) is ≠ 2 for all sufficiently large coefficients s, t.

We assume, without losing generality, that X = H and write Y = αH + βP + γQ. Since X and Y do not belong to the same Borel algebra, we must have βγ ≠ 0. Now the equation trace(exp(sX) exp(tY)) = 2 boils down to e^{s}a + e^{−s}d = 2, which means that e^{s} satisfies the quadratic equation ax^{2} −2x+d = 0. If t is sufficiently large to ensure that a ≠ 0 then the solutions of this quadratic equation are

(∗∗)  \frac{1 ± \sqrt{1 − ad}}{a} = \frac{1 ± \sqrt{−bc}}{a} = \frac{1 ± sinh(t)\sqrt{−βγ}}{cosh(t) + α sinh(t)}

and these are real only if βγ < 0. But the expressions on the right of (∗∗) remain bounded if t → ∞, so if s and t are chooser sufficiently large then e^{s} cannot satisfy the above quadratic equation and the assertion follows.

Assertion (ii) is a consequence of (i) and 8.3.

8.6. Corollary. Let X, Y ∈ \overline{Kill^{+}} and let S be the semigroup generated by x = exp(X) and y = exp(Y). Then the following assertions are equivalent:

(i) The semigroup S is contained in \overline{exp(Kill^{+})}.

(ii) There exists a Lie semialgebra in \overline{Kill^{+}} which contains both X and Y . Equivalently, either X and Y are linearly dependent or S is contained in a conjugate of Sl(2, ℝ)^{+} .

(iii) The line segment between X and Y does not meet the interior Kill^{−} of

the light cone Kill^{0} . Proof. Case 1: X and Y belong to the same Borel algebra. Then all three assertions (i)–(iii) are satisfied, so their equivalence is obvious. (Note that if X and Y are linearly independent then ℝ_{0}^{+}X + ℝ_{0}^{+}Y is a two dimensional Lie semialgebra which can be conjugated onto a Lie subsemialgebra of either ℝH + ℝ_{0}^{+}P or ℝH − ℝ_{0}^{+}P.)

Case 2: We assume that both X and Y belong to Kill^{+} but do not belong to the same Borel algebra. Then, in particular, X and Y cannot be linearly dependent. Note that every exponential subsemigroup is conjugate to a subsemigroup of Sl(2, ℝ)^{+} , by 3.8. Put

X_{0} = \frac{1}{Δ(X)}X, Y_{0} = \frac{1}{Δ(Y)}Y.

Note that X_{0}, Y_{0} sit in Hyp but not in a single Borel algebra. By 8.5(ii) our assertion (i) implies that X_{0}, Y_{0} satisfy the equivalent assertions (i)–(vi) of 8.1, conversely, if X_{0}, Y_{0} satisfy 8.1(i)–(vi) then our (i) follows. In other words, S ⊆ \overline{exp(Kill^{+})} if and only if X_{0}, Y_{0} satisfy the equivalent conditions of 8.1.

Also, our assertion (iii) holds if and only if X_{0}, Y_{0} satisfy 8.1(vi), and if 8.1(iv) is true for X_{0}, Y_{0} then our (ii) follows. Finally (ii) ⟹ (i), so the equivalence of (i)–(iii) follows.

Case 3: At least one of X, Y belongs to Kill^{0} , and X, Y do not belong to the same Borel algebra. With no loss of generality we assume that X = P and write Y = αH + βP + γQ. Note that γ ≠ 0. We first remark that in this case assertion (iii) is equivalent to γ > 0. Indeed, since by assumption α^{2} + βγ ≥ 0, the number

Δ(λX + μY) = Δ(μαH + (λ + μβ)P + μγQ) = μ^{2}(α^{2} + βγ) + λμγ

is nonnegative for all λ, μ ∈ ℝ^{+} if and only if γ > 0. Thus the set ℝ^{+}X + ℝ^{+}Y does not meet the interior Kill^{−} of the light cone if and only if γ > 0.

Next we show that (i) implies that γ > 0. Write y = \binom{a b}{c d}. Then for m ∈ ℕ we have

trace (x^{m}y) = trace (yx^{m}) = trace(y) + mc.

If γ < 0 then c < 0 and, for m sufficiently large, trace(x^{m}y) < −2, so S is not contained in the image of the exponential function, a fortiori not in \overline{exp(Kill^{+})}. It follows that (i) implies γ > 0 and hence (i) ⟹ (iii).

Since (ii) trivially implies (i) it remains to show that (iii) ⟹ (ii). Suppose that (iii) holds. Then for every n ∈ ℕ the elements X_{n} = (1 − \frac{1}{n})X + \frac{1}{n}Y , Y_{n} = \frac{1}{n}X +(1− \frac{1}{n})Y are in Kill^{+} and do not lie in the same Borel algebra, hence by the discussion of case 2 the line segment between X_{n} and Y_{n} is contained in a Lie semialgebra which lies in \overline{Kill^{+}} . Write W_{n} for the semialgebra W_{n} generated by X_{n} and Y_{n}. Then the W_{n} ’s are monotone increasing, hence the closure of the union ⋃ W_{n} is a Lie semialgebra (cf. [5], Corollary II.2.16, p. 89) which contains both X and Y . This finishes the proof.

8.7. Remark. Suppose that X and Y are linearly independent elements in Kill^{0} . Then exp(ℝ^{+}X) exp(ℝ^{+}Y) ∩ exp(Kill^{0}) = Ø.

Proof. Let us assume, with no loss of generality, that X = P, and write Y = αH + βP + γQ, with α^{2} + βγ = 0. For any s, t ∈ ℝ^{+} we have trace(exp(sX) exp(tY)) = 2 + stγ, so trace(exp(sX) exp(tY)) = 2 if and only if γ = 0, equivalently, Y is a multiple of X = P. This implies the assertion.

We now come to the main result of this section, a useful formula which expresses the diamond product ^{♢} in terms of products in Sl(2, ℝ). For investigations “in the large” it plays the role the Trotter Product Formula plays for investigations “near the identity.” This formula is based on the following observation:

8.8. Remark. Let \binom{a b}{c d} ∈ Sl(2, ℝ) and suppose that the products

ℓ(s) = ( \begin{matrix}e^{s} & 0 \\ 0 & e^{−s} \end{matrix} ) ( \begin{matrix}a & b \\ c & d \end{matrix} ) , r(s) = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ( \begin{matrix}e^{s} & 0 \\ 0 & e^{−s} \end{matrix} )

lie in \overline{exp(Kill^{+})} for certain positive reals s which can be chosen arbitrarily large. Then a > 0 (for otherwise trace(ℓ(s)) < 2 if s is large) and for any sufficiently large s ∈ ℝ^{+} the elements ℓ(s) and r(s) sit in exp(Kill^{+}). Now our formulas for the function rlog yield

(†)  rlog ℓ(s) = \frac{e^{s}a − e^{−s}d}{u} · H + \frac{2e^{s}b}{u} · P + \frac{2e^{−s}c}{u} · Q,
(‡)  rlog r(s) = \frac{e^{s}a − e^{−s}d}{u} · H + \frac{2e^{−s}b}{u} · P + \frac{2e^{s}c}{u} · Q,

where u = \sqrt{(e^{s}a + e^{−s}d)^{2} − 4}. This implies

lim_{s→∞} rlog ℓ(s) = H + \frac{2b}{a}P,  lim_{s→∞} rlog r(s) = H + \frac{2c}{a}Q.

Note that in the terminology and notation of 4.16 we have \binom{a b}{c d} · H = H + \frac{2c}{a}Q and H · \binom{a b}{c d} = H + \frac{2b}{a}P. Thus the two formulas in the display above can be considered as formulas for the natural partial left and right action of Sl(2, ℝ) on Hyp.

8.9. Theorem: An asymptotic formula for the diamond product. Assume that X, Y ∈ Hyp satisfy one (hence all) of the conditions listed in Proposition 8.1. Then X^{♢}Y is defined and

X^{♢}Y = lim_{(s,t)→(∞,∞)} rlog ( exp(sX) exp(tY))
= lim_{s→∞} lim_{t→∞} rlog ( exp(sX) exp(tY))
= lim_{t→∞} lim_{s→∞} rlog ( exp(sX) exp(tY)).

Proof. We first note that if the pair (X, Y) satisfies the conditions in 8.1 then so does (−Y, −X). Thus if we can show that under our assumptions the above three limits of rlog ( exp(sX) exp(tY)) exist and lie in hor(X) then, by applying this result to the pair (−Y, −X), we find that each of the corresponding limits of

rlog ( exp(sX) exp(tY)) = − rlog ( exp(−tY) exp(−sX))

exists and lies in − hor(−Y) = vert(Y), hence is contained in hor(X) ∩ vert(Y) = {X^{♢}Y }, and the assertion follows.

Applying a suitable inner automorphism we enforce that X = H. Pick two sufficiently large positive reals s and t. Write Y = αH + βP + γQ and exp(tY) = \binom{a b}{c d}. Then a = cosh(t) + α sinh(t), d = cosh(t) − α sinh(t), b = β sinh(t), c = γ sinh(t) and inserting this into formula (†):

rlog(exp sX exp tY) = \frac{e^{s}a − e^{−s}d}{u} · H + \frac{2e^{s}b}{u} · P + \frac{2e^{−s}c}{u} · Q,

with \sqrt{(e^{s}a + e^{−s}d)^{2} − 4}, we see that

lim_{(s,t)→(∞,∞)} rlog ( exp(sX) exp(tY))

= lim_{s→∞} lim_{t→∞} rlog ( exp(sX) exp(tY))

= lim_{t→∞} lim_{s→∞} rlog ( exp(sX) exp(tY))

= H + \frac{2β}{1 + α}P ∈ hor(H).

(note that by 8.2 we already know that 1 + α > 0) and the assertion follows.

8.10. Remark.

(i) The arguments of the above proof can be used almost verbatim to show that under the hypothesis of 8.9 the following slightly more general equality holds:

X^{♢}Y = lim_{{(s,t)→(∞,∞) \atop (X′,Y ′)→(X,Y )}} rlog ( exp(sX′) exp(tY′)).

Details are left to the reader.

(ii) For X = H, Y = αH + βP + γQ ∈ Hyp with α > −1, the proof of Theorem 8.9 also yields, without taking recourse to 4.5, the special formula H^{♢}(αH + βP + γQ) = H + \frac{2β}{1+α}P.

8.11. Corollary. Assume that X, Y ∈ Hyp satisfy one (hence all) of the conditions listed in Proposition 8.1. Let s, t be arbitrary positive reals and denote by S the semigroup generated by exp(sX) and exp(tY). Then S ⊆ exp(Kill^{+}), moreover, the diamond products X^{♢}Y and Y ^{♢}X exist and lie in the closure of the set rlog(S).

Proof. This is an immediate consequence of Theorem 8.9.

8.12. Remark. The formulas of Theorem 8.9 cannot be extended to the closure of the set of all pairs (X, Y) satisfying the conditions of Proposition 8.1. The boundary of this ‘admissible’ set consists exactly of the pairs (X, Y) with Y ∈ hor(−X) ∪ vert(−X), which means that at least one of the two products X^{♢}Y and Y ^{♢}X does not exist. For instance, if Y ∈ hor(−X) then there are arbitrarily large numbers s, t ∈ ℝ^{+} such that exp(sX) exp(tY) is unipotent, and thus rlog(exp(sX) exp(tY)) is not defined.

As an example consider the case X = H −2Q, Y = H +2P. Here X^{♢}Y is not defined (cf. 4.15), whereas Y ^{♢}X = H. The elements X and Y span a Borel subalgebra, which does not contain H. Thus no formula involving only products of the form exp(sX) or exp(tY) can yield Y ^{♢}X.

9. Rectangular bands coming from subsemigroups of Sl(2, ℝ)

9.1. Asymptotic rectangular bands. For a subset S of Sl(2, ℝ) we write

U_{0}(S) =^{def} {X ∈ Hyp | exp(ℝ^{+}X) ∩ S ≠ Ø} = rlog(S ∩ exp(Kill^{+})),
U(S) =^{def} \overline{U_{0}(S)}.

If S is a subsemigroup and contained in \overline{exp(Kill^{+})} and if U_{0}(S) is not contained in a single Borel algebra then

Asy(S) =^{def} \overline{c(U(S))} = \overline{c(U_{0}(S))} ⊂ Hyp^{−}

is called the asymptotic rectangular band of S.

This is an ambitious definition, the term ‘rectangular band’ and the epitheton ornans ‘asymptotic’ have yet to be justified. We need some preparation.

9.2. Remark. Note that for any subset S of Sl(2, ℝ) we have U_{0}(\overline{S}) = rlog(\overline{S} ∩ exp(Kill^{+})) ⊆ \overline{U_{0}(S)} = U(S), and therefore U(S) = U(\overline{S}).

9.3. Proposition. Suppose that X, Y ∈ Hyp with Y ∈ − hor(X) \ {−X}. Consider two arbitrary positive reals s, t and denote by S the semigroup generated by exp(sX) and exp(tY). Then Asy(S) contains the nilpotency point (hor(X), − hor(X)).

Proof. Without losing generality we assume that X = H and Y = −H + βP, where β ∈ ℝ \ {0}. Let m, n be natural numbers so that ms > nt. Using the formulas for exp and rlog we compute

Z(m, n) := rlog(exp(msX) exp(ntY)) = H + σP with σ = \frac{(e^{2nt} − 1)β}{1 − e^{−2ms+2nt}} .

Note that hor(Z(m, n)) = H + ℝP = hor(X). Choose now sequences ⟨m_{k}⟩ and ⟨n_{k}⟩ of natural numbers such that lim n_{k} = lim(m_{k}s − n_{k}t) = ∞. Then all elements of the sequence ⟨Z(m_{k}, n_{k})⟩ lie on the horizontal line hor(X), but no subsequence of ⟨Z(m_{k}, n_{k})⟩ converges in Hyp. Thus (cf. 4.9(iii)) we must have lim c(Z(m_{k}, n_{k})) = (hor(X), − hor(X)), which finishes the proof.

Now here is the justification for introducing the terminology of 9.1.

9.4. Theorem. Let S ⊆ \overline{exp(Kill^{+})} be a subsemigroup of Sl(2, ℝ). Suppose that U(S) is not contained in a single Borel subalgebra of 𝔰𝔩(2, ℝ). Then the following assertions hold:

(i) The subset U(S) is a full partial subsemigroup of Hyp, that is, if X, Y ∈ U(S) and X^{♢}Y is defined then X^{♢}Y ∈ U(S).

(ii) Asy(S) is a subsemigroup of the rectangular band Hyp^{−}.

(iii) Let I be a two sided ideal of S. Then U(S) = U(I) and Asy(S) =

Asy(I). Proof. Pick X, Y ∈ U_{0}(S).

Case 1: X and Y do not belong to the same Borel algebra. Then by 8.5(ii) the equivalent conditions of 8.1 are satisfied and hence by Corollary 8.11 the diamond product X^{♢}Y exists and lies in U(S). Moreover, if either X or Y belongs to U_{0}(I) then X^{♢}Y ∈ U(I), in view of 8.9.

Case 2: X and Y belong to the same horizontal or vertical line. Then X^{♢}Y = X or X^{♢}Y = Y and hence X^{♢}Y ∈ U(S), trivially. Furthermore, the line segment joining X and Y does not intersect Kill^{0} so the asymptotic formula of Theorem 8.9 shows that X^{♢}Y ∈ U(I) whenever X or Y lies in U_{0}(I).

Case 3: X and Y belong to the same Borel algebra 𝔟 but Y ∈ − hor(X). Then the diamond product X^{♢}Y does not exist and the product c(X)c(Y) is the nilpotency point (hor(X), − hor(X)). If Y ≠ −X then Proposition 9.3 shows that (hor(X), − hor(X)) ∈ Asy(S). Also, if X or Y belongs to U_{0}(I) then 9.3 implies that (hor(X), − hor(X)) ∈ Asy(I).

Thus we may assume that Y = −X. This means that there are numbers s, t ∈ ℝ^{+} such that exp(sX) ∈ S and exp(−tX) ∈ S. If s, t are rationally dependent then we may assume s = t. If s, t are independent over the rationals then the set ℕs − ℕt is dense in ℝ and therefore exp(−sX) ∈ \overline{S}.

For simplicity we suppose, without losing generality, that X = H. Since U_{0}(S) is not contained in a single Borel algebra there exists an element Z ∈ U_{0}(S) \ (ℝH + ℝQ), say Z = αH + βP + γQ with β ≠ 0. Pick ζ ∈ ℝ^{+} with exp(ζZ) ∈ S. Then for every n ∈ ℕ the element exp(nsH) exp(ζZ) exp(−nsH) lies in exp(Kill^{+}) ∩ \overline{S}, hence

Z_{n} =^{def} rlog(exp(nsH) exp(ζZ) exp(−nsH)) = e^{ns·adH}Z
= αH + e^{2ns}βP + e^{−2ns}γQ

belongs to U(S). Now lim_{n→∞} \frac{1}{βe^{2ns}} Z_{n} = P, so Proposition 4.10 shows that lim c(Z_{n}) = (hor(H), − hor(H)) ∈ Asy(S). Moreover, if either X ∈ U_{0}(I) or Y ∈ U_{0}(I) then Z_{n} ∈ U(I), for all n, hence lim c(Z_{n}) = (hor(H), − hor(H)) ∈ Asy(I).

Case 4: X and Y span a Borel algebra 𝔟 but X ∈ − hor(Y), with Y ≠ −X. Then X^{♢}Y exists. Here we assume, with no loss of generality, that Y = H. Then X can be written in the form X = −H+pP with p ≠ 0. We claim that there exists an element Z ∈ U_{0}(S) with Z ∉ ℝH + ℝP and Z ∉ −H + ℝQ. Indeed, since U_{0}(S) is not contained in ℝH + ℝP we can find a Z ∈ U_{0}(S), say Z = αH + βP + γQ with γ ≠ 0. Let s, t ∈ ℝ^{+} with exp(tZ) ∈ S, exp(sH) ∈ S. Recalling formula (‡) of Remark 8.8 we find

lim_{s→∞} rlog(exp(tZ) exp(sH)) = H + \frac{2γ sinh(t)}{cosh(t) + α sinh(t)}Q.

Replacing Z by rlog(exp(tZ) exp(sH)) for suitably large s we now enforce that Z ∉ −H + ℝQ, since γ ≠ 0 we also have Z ∉ ℝH + ℝP. Thus, applying what we have learned in Case 1 and Case 2, we see that Z^{♢}Y ∈ U(S). Note that with Z = αH + βP + γQ we have Z^{♢}Y = H + \frac{2γ}{1+α}Q. Let ⟨Y_{n} = α_{n}H + β_{n}P + γ_{n}Q⟩ be a sequence in U_{0}(S) with lim Y_{n} = Z^{♢}Y . Then α_{n} → 1, β_{n} → 0, γ_{n} ↛ 0.

We claim that X and Y_{n}, for sufficiently large n, satisfy the condition (v) of 8.1, that is, Y_{n} ∉ − hor(X) and Y_{n} ∉ − vert(X). Since − vert(X) = hor(−X) = H + ℝP we see that Y_{n} ∉ − vert(X), for large n. Also, the formula in 4.5 yields − hor(X) = H − pP + ℝ(−2pH + p^{2}P − 4Q). Thus Y_{n} ∈ − hor(X) if and only if the equations

α_{n} = 1 − 2px, β_{n} = −p + p^{2}x, γ_{n} = −4x

have a solution x ∈ ℝ. But since α_{n} → 1 and γ_{n} ↛ 0 such a solution cannot exist for all large n.

It follows that we can apply 8.11 once more, and conclude that X^{♢}Y_{n} ∈ U(S), so X^{♢}Y = lim_{n} X^{♢}Y_{n} ∈ U(S).

Furthermore, if X ∈ U_{0}(I) then, by the discussions in Case 1 and Case 2, the diamond products X^{♢}Y_{n} lie in U(I), hence X^{♢}Y ∈ U(I). Similarly, if Y ∈ U_{0}(I) then, invoking the results in Case 1 and Case 2, we see that the diamond product Z^{♢}Y lies in U(I). Hence we can choose Y_{n} ∈ U_{0}(I) and therefore X^{♢}Y = lim X^{♢}Y_{n} ∈ U(I).

Summarizing the results established case by case above we now note that for fixed X, Y ∈ U_{0}(S) the diamond product X^{♢}Y belongs to U(S) if it exists and that c(X)c(Y) ∈ Asy(S) otherwise. Passing to limits, this proves (i) and (ii). In the same vein we note that X^{♢}Y ∈ U(I) if X^{♢}Y exists and either X or Y lies in U_{0}(I); if X^{♢}Y does not exist then c(X)c(Y) ∈ Asy(I). Passage to limits shows that Asy(I) is a two sided ideal of Asy(S). But the rectangular bands are simple semigroups (by 4.12), so we conclude Asy(S) = Asy(I), and whence U(S) = c^{−1}(Asy(S)) = c^{−1}(Asy(I)) = U(I).

9.5. Remark. In the course of the above proof we also have seen that, under the assumptions of the theorem, the set

{X^{♢}Y | X, Y ∈ U_{0}(S) and X, Y satisfy the conditions of 8.1}

is dense in U(S).

9.6. Proposition. Let S ⊆ \overline{exp(Kill^{+})} be a subsemigroup of Sl(2, ℝ). Suppose that the set U_{0}(S) contains at least two elements and that it satisfies one of the following conditions:

(i) U_{0}(S) is contained in a horizontal line,

(ii) U_{0}(S) is contained in a vertical line,

(iii) U_{0}(S) is not contained in a Borel algebra.

Then U(S) is a perfect set, i.e., none of its points is isolated.

Proof. Consider an arbitrary element X ∈ U_{0}(S). Each of the conditions (i), (ii), and (iii) implies the existence of an element Y ∈ U_{0}(S) \ {X} such that X and Y satisfy one (hence all) of the conditions listed in 8.1. (In particular, the semigroup generated by exp(ℝ^{+}X) ∪ exp(ℝ^{+}Y) is a subset of exp(Kill^{+}).) Without any loss of generality let us assume that X = H. Put Y = αH + βP + γQ. Since X ≠ Y we can also assume that β ≠ 0. (The case γ ≠ 0 is treated analogously.) Fix a positive t such that exp(tY) = \binom{a b}{c d} ∈ S and a ≠ 0. Pick positive reals s and s′ such that exp(sH) and exp(s′H) lie in S. Then rlog(exp(sH) exp(tY) exp(s′H)) ∈ U_{0}(S). Furthermore, applying formula (†) of 8.8 we obtain that

lim_{s′→∞} lim_{s→∞} rlog(exp(sH) exp(tY) exp(s′H)) = lim_{s′→∞} (H + \frac{2be^{−2s′}}{a} P ) = H.

Since b ≠ 0, the matrix H cannot be an isolated point of U(S), which finishes the proof.

9.7. Example. The semigroup

Sl(2, ℕ_{0}) =^{def} \{\binom{a b}{c d} | a, b, c, d ∈ ℕ ∪ \{0\}, ad − bc = 1\}

is a discrete submonoid of Sl(2, ℝ)^{+} , it is the compression semigroup in Sl(2, ℝ) of the lattice ℕ × ℕ. We show that

(i) Sl(2, ℕ_{0})\{1} is a free semigroup, generated by the matrices x = exp(P) and y = exp(Q);

(ii) U(Sl(2, ℕ_{0})) = 𝔰𝔩(2, ℝ)^{+} ∩ Hyp.

Remark. It is well known that the group generated by x and y is Sl(2, ℤ), which is not a free group (it contains the elements \binom{0 1}{−1 0} and \binom{−1 1}{−1 0} of order 4 and 3, respectively.)

Proof. (i) Write F (x, y) for the free semigroup in two generators, and write f: F (x, y) → Sl(2, ℕ_{0}), w(x, y) ↦ w(x, y), for the canonical evaluation map. Let 1 ≠ s = \binom{a b}{c d} ∈ Sl(2, ℕ_{0}). We have to show that s = f(w) for some uniquely defined word w ∈ F (x, y). We use induction on the number n = bc. If bc = 0 then a = d = 1 and either b = 0 or c = 0, so either s = x^{b} or s = y^{c} and the assertion is obvious. Suppose now that the assertion is true for all matrices s′ = \binom{a′ b′}{c′ d′} ∈ Sl(2, ℕ_{0}) with 0 ≤ b′c′ < bc.

Case 1: a ≤ b. Then ad = 1 + bc ≤ bd hence 1 ≤ b(d − c) and thus c < d. It follows that for a ≤ b we have

sx^{−1} = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ( \begin{matrix}1 & −1 \\ 0 & 1 \end{matrix} ) = ( \begin{matrix}a & b − a \\ c & d − c \end{matrix} ) ∈ Sl(2, ℕ_{0}),

whereas

sy^{−1} = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ( \begin{matrix}1 & 0 \\ −1 & 1 \end{matrix} ) = ( \begin{matrix}a − b & b \\ c − d & d \end{matrix} ) ∉ Sl(2, ℕ_{0}).

By the induction hypothesis there is a unique word w_{1} with f(w_{1}) = sx^{−1} . We conclude that f(w) = s for w = w_{1}x. This representation is unique since there is no word w′ ∈ F (x, y) with f(w′)y = s.

Case 2: a > b. Then ad = 1 + bc > bd and therefore 1 > b(d − c), so 0 ≥ b(d − c). Since bc ≠ 0 we conclude that d ≤ c, similar to case 1 this implies

sy^{−1} = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ( \begin{matrix}1 & 0 \\ −1 & 1 \end{matrix} ) = ( \begin{matrix}a − b & b \\ c − d & d \end{matrix} ) ∈ Sl(2, ℕ_{0}).

whereas

sx^{−1} = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ( \begin{matrix}1 & −1 \\ 0 & 1 \end{matrix} ) = ( \begin{matrix}a & b − a \\ c & d − c \end{matrix} ) ∉ Sl(2, ℕ_{0}).

As in case 1 it follows that there is a unique word w with f(w) = s.

(ii) Let ρ be a positive rational number. Then there are relatively prime positive integers p, q such that ρ = \frac{p}{q}. Remembering elementary number theory we find infinitely many pairs m, n ∈ ℕ such that pm − qn = 1. The quotients \frac{n}{m} of such pairs (m, n) converge, and lim \frac{n}{m} = lim \frac{pm−1}{qm} = \frac{p}{q} = ρ. Furthermore,

rlog ( \begin{matrix}m & n \\ q & p \end{matrix} ) = \frac{m − p}{\sqrt{(p + m)^{2} − 4}}H + \frac{2n}{\sqrt{(p + m)^{2} − 4}}P + \frac{2q}{\sqrt{(p + m)^{2} − 4}}Q

and for m → ∞ this matrix converges to H + 2ρP. Thus, by continuity, H + ℝ_{0}^{+}P ⊂ U(Sl(2, ℕ_{0})). In the same way, or using the anti-automorphism s ↦ s^{T} we see that H + ℝ_{0}^{+}Q ⊂ U(Sl(2, ℕ_{0})). Similarly, we find

rlog ( \begin{matrix}p & n \\ q & m \end{matrix} ) = \frac{p − m}{\sqrt{(p + m)^{2} − 4}}H + \frac{2n}{\sqrt{(p + m)^{2} − 4}}P + \frac{2q}{\sqrt{(p + m)^{2} − 4}}Q

and we see that −H + ℝ_{0}^{+}P and −H + ℝ_{0}^{+}Q lie in U(Sl(2, ℕ_{0})). Because of the rectangular structure of U(Sl(2, ℕ_{0})) this implies the assertion.

9.8. Example. The semigroup S = Sl(2, ℕ_{0}) ∩ S_{1}^{1} , where S_{1}^{1} is defined as in 6.12, is a discrete subsemigroup of Sl(2, ℕ_{0}). Moreover, it can be checked easily that for nonnegative integers a, b, c, d with ad − bc = 1 the following assertions (i)–(iii) are equivalent:

(i) a + b ≥ c + d, and a + c ≥ b + d;

(ii) (a + b)(a − c) ≥ 1, and (a + c)(a − b) ≥ 1;

(iii) a > b, and a > c.

Moreover, if a = d then these inequalities imply b = c, hence a^{2} = b^{2} + 1, which means a = 1, b = 0. A straightforward calculation, along the lines of the preceding example, shows that for 1 ≠ x ∈ Sl(2, ℕ_{0}) we have xS ∩ S ≠ Ø if and only if x ∈ \binom{1 1}{0 1}Sl(2, ℕ_{0}), and similarly, Sx ∩ S ≠ Ø if and only if x ∈ Sl(2, ℕ_{0})\binom{1 0}{1 1}. Thus it follows that

S = \{1\} ∪ \binom{1 1}{0 1}Sl(2, ℕ_{0})\binom{1 0}{1 1}.

Also, by a well known criterion (cf., e.g., [3] II, Corollary 9.8, p.119), this implies that S \ {1} is a free semigroup. (It can be shown that S has infinitely many generators.) Recall from 6.12 that the Lie wedge W_{1}^{1} of S_{1}^{1} is spanned by the four vectors H, H + 2P, H + 2Q, 2(P + Q). By the preceding example we know that U_{0}(Sl(2, ℕ_{0})) is dense in 𝔰𝔩(2, ℝ)^{+} ∩ Hyp, so we conclude that U_{0}(S) is dense in W_{1}^{1} ∩ Hyp, hence U(S) = W_{1}^{1} ∩ Hyp.

10. Rectangular domains and umbrella sets

10.1. The umbrella set of a subset in a Lie group. Let S be a subset of a Lie group G with Lie algebra 𝔤. Then we define the umbrella set Umb(S) of S as:

Umb(S) =^{def} {X ∈ 𝔤 | ∃t_{0} > 0 : exp(tX) ∈ S, ∀t > t_{0}}.

In the present notes S will always be a subsemigroup of G.

The term ‘umbrella set’ is derived from our situation in Sl(2, ℝ). If S is open and S ≠ Sl(2, ℝ) then Umb(S) = ℝ^{+}U_{0}(S), so Umb(S) is formed by ‘unfolding’ the set U_{0}(S) along the half lines passing through its points, like a ‘knirps.’ For general semigroups S the set Umb(S) is not fit to serve as an asymptotic object, but its closure \overline{Umb}(S) is.

10.2. Remark. The above definition provides a variant of Lawson’s ‘horizon’ concept ([10], p.21). If S is a subsemigroup with nonempty interior of a Lie group G then Lawson defines the horizon of S as the set

L_{∞}(S) =^{def} {X ∈ 𝔤 | exp(tX) ∈ Int S for some t > 0}.

It is not difficult to see that L_{∞}(S) = Umb(Int S) (see the discussion below). This observation as well as other elementary properties of Umb(S) in the case of open semigroups S are contained (at least implicitly) in [10]. The major reason for our slight deviation from Lawson’s concept is that we are interested also in one parameter subsemigroups meeting boundary points of S and in semigroups S with Int S = Ø.

10.3. Properties of Umb(S).

(i) Obviously Umb(S) is closed under multiplication with positive scalars, but except for the case [X, Y] = 0 it is not clear from the definition that X, Y ∈ Umb(S) implies that X + Y ∈ Umb(S). In fact, examples show that in the general context Umb(S) need not be additively closed, even if S is open and G is nilpotent (cf. [11],[12]). Also, if S is closed then Umb(S) need not be a closed subset of 𝔤. To see this, consider, for example, the subsemigroup T = {(x, y) ∈ ℝ^{2} | y ≥ 1} of ℝ^{2} . A simple computation shows that Umb(T) = {(x, y) ∈ ℝ^{2} | y > 0}.

(ii) Suppose that exp(X) ∈ Int S. Then there exist a neighborhood U of X in 𝔤 and a positive number T such that exp(tY) ∈ Int S for all t ≥ T and all Y ∈ U. In particular, U ⊆ Umb(Int S). This also shows that Umb(S) is open whenever S is open.

Proof of (ii): Since exp is continuous there is an ε > 0 and a neighborhood U of X in 𝔤 such that exp([1, 1 + ε] · U) ⊆ Int S. Since Int S is a subsemigroup, we have exp([n, n(1 + ε)] · Y) ⊆ Int S for each n ∈ ℕ and each Y ∈ U. But if n ∈ ℕ with n ≥ \frac{1}{ε} then exp([n, n + 1]) · U) ⊆ Int S, so the assertion holds with T = \frac{1}{ε} + 1.

(iii) If S is an open subsemigroup of Sl(2, ℝ) and S ≠ Sl(2, ℝ) then Umb(S) = ℝ^{+}U_{0}(S).

We will now show that in the case of G = Sl(2, ℝ) umbrella sets of subsemigroups have “nice” properties. We start with the remarkable fact that for open connected subsemigroups S the set U_{0}(S) is a rectangular domain.

10.4. Theorem. Let S be an open connected subsemigroup of Sl(2, ℝ) with S ≠ Sl(2, ℝ). Then S ⊆ exp(Kill^{+}) and D = U_{0}(S) is a rectangular domain.

Proof. The inclusion S ⊆ exp(Kill^{+}) follows from 3.4(iii). Also, D is open, since S is open and exp is a homeomorphism on Kill^{+} , and D is connected, since S is connected and rlog is continuous. We know already from 9.4 that the closure \overline{c(D)} of c(D) in Hyp^{−} is a subsemigroup of the rectangular band Hyp^{−}. Thus \overline{c(D)} = \overline{hor(D)} × \overline{vert(D)}. Furthermore, 8.9 implies that the diamond product X^{♢}Y exists for every X, Y ∈ D, hence \overline{hor(D)} and \overline{vert(D)} are two proper arcs. We conclude that Int (\overline{c(D)}) = hor(D) × vert(D), so Int (\overline{c(D)}) contains no nilpotency points. This yields that Int(\overline{D}) = c^{−1}( Int (\overline{c(D)})) is a ♢ -semigroup.

Thus all we need to show is that D is the interior of \overline{D}. Suppose that there exists an interior point of \overline{D} which does not belong to D. Applying a suitable inner automorphism of 𝔰𝔩(2, ℝ) we enforce that this point is H.

Since H is an interior point of \overline{D} the horizontal line hor(H) = H + ℝP dissects \overline{D}, that is to say, \overline{D} \ H + ℝP is the union of two disjoint nonvoid open subsets. Since D is connected and dense in \overline{D} this means that D ∩ hor(H) ≠ Ø. Similarly, D ∩ vert(H) ≠ Ø. Thus there exist points X and Y in D with X^{♢}Y = H, we write X = H + βP, Y = H + γQ. We claim that X and Y can be chooser so that β and γ have the same sign.

Indeed, if this is not the case then there exists a nonzero real number β such that H + βP ∈ D but H − ℝ^{+}βP ∩ D = H + ℝ^{+}βQ ∩ D = Ø. This means, however, that D does not meet the union H − ℝ^{+}βP ∪ {H} ∪ H + ℝ^{+}βQ. Since the latter set also dissects \overline{D} we therefore arrive at a contradiction.

The inner automorphism φ = e^{ad tH}, with t = \frac{1}{4}(log |γ|−log |β|) satisfies φ(H) = H, φ(|β|P) = \sqrt{βγ}P and φ(|γ|Q) = \sqrt{βγ}Q. Applying φ we thus enforce that β = γ, so Y is the transpose of X. Since D is open the matrix

X_{ε} = X + ε(H + \frac{β}{2} P − \frac{2}{β} Q) = (1 + ε)H + (1 + \frac{ε}{2})βP − ε\frac{2}{β} Q

and its transpose

Y_{ε} = Y + ε(H − \frac{2}{β} P + \frac{β}{2} Q) = (1 + ε)H − ε\frac{2}{β} P + (1 + \frac{ε}{2})βQ

are also in D for all sufficiently small ε ≥ 0. We know from 10.3 that there exist positive reals t_{0} and ε_{0} such that exp(tX_{ε}) and exp(tY_{ε}) lie in S for t ≥ t_{0} and ε ∈ [0, ε_{0}]. We now fix t ≥ t_{0} such that \frac{cosh(t)}{sinh(t)} − 1 < ε_{0} . A straightforward calculation using our formulas for exp shows that

exp(tX_{ε}) exp(tY_{ε}) = ( \begin{matrix}a & b \\ b & d \end{matrix} ) , where

b = β sinh^{2}(t)\{\frac{−2ε}{β^{2}} (\frac{cosh(t)}{sinh(t)} + 1 + ε) + (1 + \frac{ε}{2}) (\frac{cosh(t)}{sinh(t)} − 1 − ε)\}.

If in this formula we put ε = 0 then b and β have the same sign, whereas for ε = \frac{cosh(t)}{sinh(t)} − 1 these numbers have the opposite sign. It follows that b = 0

for some ε in the interval (0, \frac{cosh(t)}{sinh(t)} − 1), so exp ℝ^{+}H meets S and therefore H ∈ D, a contradiction. This finishes our proof.

10.5. Theorem. Let S ≠ Sl(2, ℝ) be a subsemigroup of Sl(2, ℝ) with dense and connected interior. Then the following assertions hold:

(i) \overline{Umb}(S) is a Lie semialgebra and its exponential image exp(\overline{Umb}(S)) is a closed semigroup;

(ii) Umb(Int S) is the algebraic interior of \overline{Umb}(S); (iii) The set S_{∗} = {s ∈ Sl(2, ℝ) | s^{n} ∈ Int S for some power n ∈ ℕ} is an

open semigroup. More specifically, we have

S_{∗} = exp(Umb(Int S)) ∪ − exp(Umb(Int S)),
\overline{S_{∗}} = \overline{\{s ∈ Sl(2, ℝ) | 1 ≠ s^{n} ∈ S for some power n ∈ ℕ\}}.

Proof. We first note that S ⊆ exp(\overline{Kill^{+}}), by 3.4(iii).

(i) Since S has dense interior we see that Umb(Int S) is dense in \overline{Umb}(S). Furthermore, Umb(Int S) = ℝ^{+} rlog(Int S) by 10.3(iii), and rlog(Int S) is a rectangular domain by 10.4. Combining this with 5.5(i) we get assertion (i).

(ii) Since for a convex open subset C of a finite dimensional vector space we always have Int(\overline{C}) = C it follows that Int(\overline{Umb}(S)) = Int(\overline{Umb}(Int S)) = Umb(Int S).

(iii) We know from (i) and (ii) that exp(Umb(Int S))∪− exp(Umb(Int S)) is an open semigroup. Thus we are left to show that the two equalities hold. In order to prove the first one it suffices to show the inclusion S_{∗} ⊆ exp(Umb(Int S))∪ − exp(Umb(Int S)). Pick s ∈ S_{∗}, and let n ∈ ℕ with s^{n} ∈ Int S. We suppose that s ∈ exp(𝔰𝔩(2, ℝ)). By 2.2(ii) and since (s ∈ S_{∗}) ⟺ (−s ∈ S_{∗}), this will inflict no loss of generality. Write s = exp(X). Then s^{n} = exp(nX) ∈ Int S, so by 10.3(ii) X ∈ Umb(Int S), hence s = exp X ∈ exp(Umb(Int S)).

To show the second identity we only have to prove that

\{s ∈ Sl(2, ℝ) | 1 ≠ s^{n} ∈ S for some power n ∈ ℕ\} ⊆ \overline{S_{∗}}.

Pick s ∈ Sl(2, ℝ) such that 1 ≠ s^{n} ∈ S for a suitable n ∈ ℕ. Observe that (−s)^{2n} = s^{2n} also lies in S \ {1}. As before we therefore may and do suppose that s = exp(X) with 0 ≠ X ∈ 𝔰𝔩(2, ℝ). Now X cannot lie in Kill^{−} (since S ⊂ exp(\overline{Kill^{+}}) and s^{n} ≠ 1), so s = exp(X) ∈ exp(\overline{Kill^{+}}). Let U be an open neighborhood of s in exp(\overline{Kill^{+}}). Define p_{n}: exp(\overline{Kill^{+}}) → exp(\overline{Kill^{+}}), x ↦ x^{n}. Since the restriction of exp to \overline{Kill^{+}} is a homeomorphism, the map p_{n} is a homeomorphism, and therefore p_{n}(U) is an open neighborhood of s^{n} in exp(\overline{Kill^{+}}). Since Int S is dense in S we therefore find an element s_{∗} ∈ U with p_{n}(s_{∗}) ∈ Int S. By definition, s_{∗} ∈ U ∩ S_{∗} and we conclude that s_{∗} ∈ \overline{S_{∗}}. The assertion follows.

10.6. Corollary. Let S be an open subsemigroup of Sl(2, ℝ) with S ≠ Sl(2, ℝ). Then there are countably or finitely many Lie semialgebras W_{j}, j ∈ J, with nonvoid interior and contained in \overline{Kill^{+}} , such that

Umb(S) = ⋃_{j∈J} Int W_{j}.

Proof. Note first that Umb(S) = Umb(S ∩ exp(Kill^{+})). By Proposition 3.5 we know that S ∩ exp(Kill^{+}) is the union of the semigroups

S_{c}(x) = ⋃_{k∈ℕ} C(x^{k}), x ∈ S ∩ exp(Kill^{+}),

where C(x) denotes the connected component of x in S. (Since S is open, the family {C(x)} is countable or finite.) Furthermore, for every x ∈ S ∩ exp(Kill^{+}) there is an index n such that Umb(S_{c}(x)) = Umb(C(x^{n})). Now our assertion is an immediate consequence of Theorem 10.5.

10.7. Remark. Let S be a subsemigroup of Sl(2, ℝ) with dense interior and assume that S ⊂ \overline{exp(Kill^{+})}. Then \overline{Umb}(S) = \overline{Umb}(I) for every two sided ideal I in S. This can be seen from the following chain of inclusions and identities (see also Theorem 9.4)

\overline{Umb}(S) ⊇ \overline{Umb}(I) ⊇ \overline{Umb}(Int I) = \overline{ℝ^{+}U_{0}(Int I)} = \overline{ℝ^{+}U(Int I)}
= \overline{ℝ^{+}U(S)} = \overline{Umb}(S).

Further generalizations to arbitrary subsemigroups S ≠ Sl(2, ℝ) with dense interior are left to the reader.

10.8. Theorem. Let S be a Lie subsemigroup of Sl(2, ℝ) with non-empty interior and let W be its Lie wedge. Then \overline{Umb}(S) is the semialgebra generated by W, i.e.,

\overline{Umb}(S) = ⋂\{W_{1} ⊆ 𝔰𝔩(2, ℝ) | W ⊆ W_{1} and W_{1} is a semialgebra\}.

Proof. Denote the intersection

⋂{W_{1} ⊆ 𝔰𝔩(2, ℝ) | W ⊆ W_{1} and W_{1} is a semialgebra}

by W̃. Obviously, Umb(S) contains W. We know by Theorem 10.5 that \overline{Umb}(S) is a semialgebra. Thus W̃ ⊆ \overline{Umb}(S). Since W̃ is a Lie semialgebra contained in \overline{Kill^{+}} , its exponential image exp W̃ is an exponential subsemigroup of Sl(2, ℝ). On the other hand, the inclusion W ⊆ W̃ implies that S ⊆ exp(W̃), hence Umb(S) ⊆ Umb(exp(W̃)) = W̃. Thus \overline{Umb}(S) ⊆ W̃.

10.9. Remark. The Lie semialgebra V generated by a wedge W with W ⊆ \overline{Kill^{+}} , and not contained in a single Borel algebra, can be computed comfortably as the intersection of all closed half spaces bounded by a Borel algebra which contain W. Alternatively: V = \overline{ℝ^{+} · M}, where

M = {X^{♢}Y | X, Y ∈ W ∩ Hyp and both X^{♢}Y and Y ^{♢}X exist}.

To see this, notice that the algebraic interior W_{0} of W lies in Kill^{+} and that W_{0} ∩ Hyp is connected, hence D = (W_{0} ∩ Hyp)^{♢}(W_{0} ∩ Hyp) is a rectangular domain. By 5.5(i) we know that \overline{ℝ^{+}D} is a Lie semialgebra. Since it contains W as well as M this implies the assertion.

10.10. Example. Let W be the wedge spanned by the three vectors A = H + 2P, B = H + 2Q, and H. Then W is not a semialgebra, since [A, B] = 4H − 4P − 4Q does not belong to the span of A, B. The Lie semialgebra generated by W is spanned by the four vectors A, B, H, and B^{♢}A = P + Q. This semialgebra is the intersection of 𝔰𝔩(2, ℝ)^{+} with its conjugate φ(𝔰𝔩(2, ℝ)^{+}), where φ is the inner automorphism which induces a rotation of 𝔰𝔩(2, ℝ)^{+} about the axis ℝ(P − Q) by π/2.

10.11. Proposition. Let W be the Lie wedge of a three dimensional Lie subsemigroup S of Sl(2, ℝ). Then W, Umb(S), and \overline{Umb}(S) contain the same nilpotent matrices:

\overline{Umb}(S) ∩ Kill^{0} = W ∩ Kill^{0} = Umb(S) ∩ Kill^{0} .

Proof. Since W ⊆ Umb(S) it suffices to show the equality \overline{Umb}(S) ∩ Kill^{0} = W ∩ Kill^{0} . Also, by [5] V.4.23 (p.418) all Lie wedges in 𝔰𝔩(2, ℝ) with nontrivial edge are conjugate to 𝔰𝔩(2, ℝ)^{+} , so our assertion is trivial if W is not pointed.

Thus let us assume that W is pointed.

Step 1. We first show that if W does not contain nonzero nilpotent elements then neither does \overline{Umb}(S). Assume W to be free of nonzero nilpotent elements. Then W \ {0} ⊆ Kill^{+} . Let g: Kill^{+} → Hyp, X ↦ (1/Δ(X))X. The set M =^{def} W ∩ Hyp = g(W \ {0}) is compact (since W \ {0} ⊆ Kill^{+} ) and connected. Since W \ {0} ⊆ Kill^{+} we see that for any two points X, Y ∈ M the line segment conv{X, Y } does not meet the light cone, so Theorem 8.9 implies that all diamond products X^{♢}Y with X, Y ∈ M exist. Using Theorem 10.8 and Remark 10.9 we conclude that \overline{Umb}(S) = \overline{ℝ_{0}^{+}(M^{♢}M)}. Since M is compact we therefore see that \overline{Umb}(S) = ℝ_{0}^{+}(M^{♢}M), and that \overline{Umb}(S) indeed contains no nonzero nilpotent elements.

Step 2. Next we suppose that \overline{Umb}(S) contains a nilpotent element N which is, however, not contained in W. To simplify our arguments we assume, not losing generality, that W ⊆ 𝔰𝔩(2, ℝ)^{+} and N = Q. By 10.9 this means that there exist elements X_{n}, Y_{n} ∈ W ∩ Hyp and numbers λ_{n} ∈ ℝ^{+} such that Q = lim λ_{n}(X_{n}^{♢}Y_{n}). Also, remembering step 1, we know that P ∈ W. By 4.10

(∗)  lim_{n} c(X_{n}^{♢}Y_{n}) = (hor(−H), − hor(−H)).

We claim that a suitable subsequence of ⟨X_{n}⟩ converges. If not then we can find a subsequence ⟨X_{m}⟩ of ⟨X_{n}⟩ and positive numbers λ_{m} with λ_{m} → 0 such that ⟨λ_{m}X_{m}⟩ converges to a nonzero element. This element must be nilpotent and is contained in W, hence lim_{m} λ_{m}X_{m} = P. Invoking 4.10 once more, we see that lim_{m} hor(X_{m}) = hor(H), a contradiction to (∗).

Thus ⟨X_{n}⟩ has a convergent subsequence ⟨X_{k}⟩, and, by the same argument, ⟨Y_{n}⟩ has a convergent subsequence ⟨Y_{k}⟩. By (∗) lim_{k} c(X_{k}, Y_{k}) = (hor(−H), vert(H)), therefore

hor(lim_{k} X_{k}) = hor(−H), vert(lim_{k} Y_{k}) = vert(H).

It follows that A = lim_{k}(X_{k} + Y_{k}) ∈ ℝQ ∩ W ⊆ ℝ_{0}^{+}Q. If A = 0 then 0 ≠ lim_{k} X_{k} = − lim_{k} Y_{k} ∈ W, a contradiction to our assumption that W is pointed. If A ≠ 0 then Q ∈ W, contrary to our assumption in step 2.

10.12. Example. We compute \overline{Umb}(S) and Umb(S) of the Lie semigroup S = Sl(2, ℝ)^{++} of Example 6.5. The Lie wedge of S is W = ℝ_{0}^{+}H +ℝ_{0}^{+}P +ℝ_{0}^{+}Q. Then Theorem 10.8 shows that \overline{Umb}(S) is the Lie semialgebra generated by W. Since W contains the linearly independent nilpotent elements P and Q we conclude that \overline{Umb}(S) = 𝔰𝔩(2, ℝ)^{+} . For \binom{a b}{c d} = exp(−αH + βP) with α > 0, β ∈ ℝ we do not have a ≥ 1, so −H + ℝP ∩ Umb(S) = Ø. Similarly, −H + ℝQ ∩ Umb(S) = Ø. Thus Umb(S) = Umb(Int S) ∪ W.

10.13. Example. Consider the Lie semigroup S = Sl(2, ℝ)^{++,+−} of Example 6.6. The Lie wedge of S is W = ℝ_{0}^{+}P + ℝ_{0}^{+}Q. Since S has interior points Theorem 10.8 applies and shows that \overline{Umb}(S) = 𝔰𝔩(2, ℝ)^{+} , which can be verified easily by direct computation as well. Thus every one parameter subsemigroup exp ℝ^{+}X with X ∈ algint(𝔰𝔩(2, ℝ)^{+}) meets the interior of S. However H ∉ Umb(S), so Umb(S) is not closed. The only upper [lower] triangular matrices in S are the unipotent elements exp(tP) [exp(tQ)], with t ≥ 0. Thus Umb(S) = Umb(Int S) ∪ W. Note that W is only two dimensional; nevertheless the asymptotic behavior of the interior of Sl(2, ℝ)^{++,+−} is the same as that of the interior of the apparently ‘larger’ semigroup Sl(2, ℝ)^{+} .

Note that, in accordance with Proposition 10.11, W and Umb(S) contain the same nilpotent elements.

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[17] Ruppert, W. A. F., and B. E. Breckner, Bohr compactifications of subsemigroups of Sl(2, ℝ), (forthcoming).

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B. Breckner Babeş-Bolyai University Faculty of Mathematics and Computer Science Str. M. Kogălniceanu 1 RO-3400 Cluj-Napoca Romania brigitte@math.ubbcluj.ro and Institut für Mathematik und Angewandte Statistik Universität für Bodenkultur Peter Jordanstr. 82 A-1190 Wien Austria breckner@edv1.boku.ac.at W. A. F. Ruppert Institut für Mathematik und Angewandte Statistik Universität für Bodenkultur Peter Jordanstr. 82 A-1190 Wien Austria ruppert@edv1.boku.ac.at

Received August 7, 2000 and in final form March 8, 2001