Journal of Lie Theory Volume 14 (2004) 73–109 \begin{matrix}c \\ ◯ \end{matrix} 2004 Heldermann Verlag

On Compactification Lattices of Subsemigroups of SL(2,ℝ)

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

Communicated by Jimmie D. Lawson

Dedicated to Karl Heinrich Hofmann on the occasion of his 70^{th} birthday

Abstract. Using the tools introduced in [2] we investigate topological semigroup compactifications of closed connected submonoids with dense interior of Sl(2,ℝ). In particular, we show that the growth of such a compactification is always contained in the minimal ideal, and describe the subspace of all minimal idempotents (typically a two-cell) and the maximal subgroups (these are always isomorphic with a compactification of ℝ). For a large class of such semigroups we give explicit constructions yielding all possible topological semigroup compactifications and determine the structure of the compactification lattice.

2000 Mathematics Subject Classification: 22E15, 22E46, 22A15, 22A25

Key words and phrases: Bohr compactification, lattice of compactifications, asymptotic homomorphism, subsemigroups of Sl(2,ℝ), Lie semigroups, Lie semialgebras, diamond product, rectangular domain, umbrella set, divisible semigroup, UDC semigroup.

1. Introduction

Let S be a topologized semigroup and κ: S → S^{κ} a continuous homomorphism into a compact (Hausdorff) topological semigroup S^{κ}. If κ(S) is dense in S^{κ} then the pair (S^{κ}, κ) is said to be a topological semigroup compactification of S. Given a class S of such semigroups S we are faced with the following basic tasks:

(i) For any semigroup S ∈ S find, up to equivalence, all topological semigroup compactifications, if possible, give explicit constructions. In particular, give a construction yielding the universal topological semigroup compactification, the Bohr compactification of S. (This universal object always exists, by the Adjoint Functor Theorem.)

∗ This author was supported by the Deutsche Forschungsgemeinschaft.

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag (ii) With every compact topological semigroup go various special objects: the set of idempotents, the maximal subgroups, the minimal ideal and its subobjects. Describe these for the compactifications of S and relate them to interesting objects connected with S and its structural features.

(iii) It is not difficult to see that the topological semigroup compactifications of a fixed semigroup S form a lattice. Describe the structure of this lattice in terms of “known” lattices.

Success in carrying out this program should benefit the structure theory as well as the harmonic analysis on S. In the following two well known examples the above questions can be answered in a very complete and satisfactory manner:

A. The class of all abelian locally compact topological groups. (Cf., e.g., Hewitt-Ross [10], p.430ff) Here we have the following simple construction: given an abelian locally compact topological group G we pick any subgroup H_{d} of the discretization (Ĝ)_{d} of the dual group Ĝ and let G^{κ} = (H_{d})ˆ . Then G^{κ} is compact and there is a natural morphism κ: G → G^{κ} with dense image. Conversely, every topological semigroup compactification of G can be obtained in this way, and the lattice of all such compactifications is isomorphic with the lattice of all subgroups of (Ĝ)_{d}. The Bohr compactification corresponds to the full group (Ĝ)_{d}. (The objects addressed in (ii) are trivial in the case of groups.)

B. The class of all convex cones in a finite dimensional real vector space. Explicit constructions yielding the Bohr compactification C^{b} of a finite dimensional (closed) cone C have been given by M. Friedberg [8,9] and, later on, by one of the authors [16]. It has been shown that the idempotents of the Bohr compactification of a finite dimensional cone C are in one-to-one correspondence with the faces of the dual cone C^{∗} and that the ‘accessible idempotents’ (those in the closure of a one parameter subsemigroup) in the growth of the compactification correspond exactly to the exposed faces of C^{∗}. We cannot enter here into the details of the compactification lattice of C, this lattice can be described in terms of the lattice of subcones of the dual cone and the compactification lattices of finite dimensional vector spaces over ℝ. To give the reader a rough idea of what is going on we only remark that the H-reduced compactifications of C are in one-one correspondence to the subcones of C^{∗}. Similar to the situation in case A the key element in the discussion is the existence of a separating family of semicharacters on C. (A general exposition of various relations between compactifications and dual objects is planned for a forthcoming paper.)

The natural nonabelian analogues of cones in real vector spaces are the closed divisible subsemigroups of connected Lie groups, these are exactly the exponential Lie semigroups. The exponential Lie semigroups are classified in the memoir [12], where also a fairly complete description of their structure is given. A reduced exponential Lie semigroup S (i.e., an exponential Lie semigroup S containing no nontrivial normal subgroup of the Lie group generated by S) can always be dissected into (1) a direct product of one or more exponential Lie semigroups living in Sl(2,ℝ) and (2) an exponential Lie semigroup living in a group which is an almost direct product of a centerfree diagonally metabelian Lie group with a covering group of a compact Lie group. The compactifications of divisible semigroups of type (2) are quite amenable, special cases such as the “affine triangle” have been studied for decades (cf., e.g., [6],[13],[14], [17]).

Thus the first step for carrying out our program (i)–(iii) for divisible subsemigroups of Lie groups is to investigate the topological semigroup compactifications of divisible subsemigroups of Sl(2,ℝ). It soon turns out, however, that for this task we need a very detailed knowledge of general structural features such as asymptotic behavior and the rectangular structure of Sl(2,ℝ) [2], perfectness and aliens [3], congruences in subsemigroups of Lie groups [4]. Also, the appropriate methods and ideas are developped best in a context as general as might be presumed, namely in the class of closed connected proper submonoids S of Sl(2,ℝ) with dense interior. In the rest of the introduction we now always assume that S belongs to this class.

The central and most general result of these notes is that for every compactification (S^{κ}, κ) of S we have (a) S^{κ} = κ(S)∪M(S^{κ}), (b) there is a continuous and surjective map ε assigning to every asymptotic direction of S (that is, to every element in the asymptotic rectangular band) a minimal idempotent of S^{κ}, and (c) all maximal subgroups of S^{κ} contain a dense homomorphic image of ℝ. If we mod out the maximal subgroups of S^{κ} then ε becomes a homomorphism, and there are only four possibilities for the kernel congruence.

In the generic case S is contained in a perfect exponential semigroup, hence is perfect itself. For this case we give an explicit construction yielding all possible topological semigroup compactifications of S, we show that the idempotents in the compactifications correspond to the asymptotic directions in S, and we describe the compactification lattice of S as the union of copies of three well known lattices: the lattice of all compactifications of ℝ, the four element diamond lattice, and the lattice formed by the empty set and the closed ideals of S.

Similar results hold if S is not perfect, in this case the compactification lattice is isomorphic with the lattice of all closed ideals containing the alien elements of S. These two cases cover already all divisible subsemigroups S.

In the remaining case, where S itself, but not its exponential hull, is perfect, we still can give fairly detailed information, but one important detail is missing: we do not know whether the above mapping ε can be injective (and if so, why). Up to now we only have examples for such semigroups where the minimal ideal of the Bohr compactification is singleton. (Nevertheless: Under a slight restriction it can be shown that every maximal subgroup in a topological semigroup compactification of such a semigroup S must be trivial.)

Acknowledgements. We want to express our gratitude to our friend and teacher Karl H. Hofmann with whom we had many enjoyable and encouraging discussions about the subject. Thanks go also to the Deutsche Forschungsgemeinschaft DFG for providing financial support to Brigitte E. Breckner.

2. A topological Rees extension

2.1. The aim of this section is to introduce a generalization of the usual one point compactification, where instead of a zero element we attach a more general compact semigroup, which acts as an ideal. (A variant of this construction was given already in [15], see 4.11 on page 35–36.) Clearly, this construction will yield a compact topological semigroup only if S is perfect, that is, its multiplication is a perfect mapping (a closed continuous map with compact fibers; cf. [7] p.236).

Since every semigroup acts on each of its ideals by left and right translations we have to consider two-sided semigroup actions.

2.2. Actions of semigroups on semigroups. Let X, Y be semigroups. Then a map α: (X × Y) ∪ (Y × X) → Y , (a, b) ↦ ab, is said to be a two-sided semigroup action of X on Y , or an s-action for short, if the associativity rules

(x_{1}x_{2})y = x_{1}(x_{2}y) and y(x_{1}x_{2}) = (yx_{1})x_{2}
x_{1}(yx_{2}) = (x_{1}y)x_{2} and (y_{1}x)y_{2} = y_{1}(xy_{2})
x(y_{1}y_{2}) = (xy_{1})y_{2} and y_{1}(y_{2}x) = (y_{1}y_{2})x

hold for any choice of x_{1}, x_{2}, x ∈ X and y_{1}, y_{2}, y ∈ Y .

If X contains an identity 1 then we require, in addition, that 1y = y1 = y, for all y ∈ Y . If X and Y are topological semigroups then our s-action will be assumed to be continuous.

Clearly, the restriction λ [μ] of α to X × Y [Y × X] is a left [right] action of X on Y .

2.3. Example. (i) Let G_{ℓ}, G_{r} be closed subgroups of a topological group G, write L for the homogeneous space G/G_{ℓ} of left cosets gG_{ℓ}, g ∈ G, and R for the homogeneous space G_{r}\G of right cosets G_{r}g, g ∈ G. Then the product space Y = L × R is a topological semigroup with respect to the rectangular multiplication (x, y)(x′, y′) = (x, y′), and the natural actions

G × Y → Y, (g, (g′G_{ℓ}, G_{r}g′′) ↦ (gg′G_{ℓ}, G_{r}g′′),
Y × G → Y, ((g′G_{ℓ}, G_{r}g′′), g) ↦ (g′G_{ℓ}, G_{r}g′′g)

combine to an s-action of G on Y (which is jointly continuous).

(ii) If X is a subsemigroup of G and Y_{1} ⊆ G/G_{ℓ}, Y_{2} ⊆ G_{r}\G with XY_{1} ⊆ Y_{1} and Y_{2}X ⊆ Y_{2} then the s-action of G on Y defined in (i) restricts naturally to an s-action of X on Y_{1} × Y_{2} .

In the next section we shall use an example of type 2.3(ii) in the case where X is a subsemigroup of G = Sl(2,ℝ), and G_{ℓ} [G_{r}] is the subgroup of all upper [lower] triangular matrices.

2.4. Remark. (i) Every s-action of X on Y defines, in the obvious way, an associative multiplication on the disjoint union Z = X ∪ Y . Pick a, b ∈ X ∪ Y . If both a and b belong to either X or Y then their product is defined by the multiplication already given on X or Y , respectively; otherwise it is defined by the s-action of X on Y . Note that in the so defined semigroup S the subset Y is an ideal.

(ii) If X and Y are topological semigroups and the s-action is continuous then the semigroup Z of (i) will be a topological semigroup if it is provided with the sum topology. In order to make Z a compact semigroup we have, however, to use a coarser topology, even if we assume that Y is compact. In the present paper we get this coarser topology with the aid of a ‘gluing map’ φ: X → Y , which is asymptotically an equivariant homomorphism. To express this general idea more precisely we need some preparations.

2.5. Notational conventions. (i) In the following all topological spaces will be assumed to be Hausdorff.

(ii) If X is a locally compact topological space, and ⟨x_{i}⟩ is a net in X, then lim x_{i} = ∞, or x_{i} → ∞ for short, will mean that ⟨x_{i}⟩ has no convergent subnet in X. In other words, if we think of X as embedded in its one point compactification X ∪ {∞} then lim x_{i} = ∞ in the usual sense.

2.6. A gluing construction for locally compact spaces. Let X, Y be (disjoint) locally compact spaces, K a compact subset of X, and let φ: X \K → Y be a continuous map.

Then we denote with X ⊔_{φ} Y the topological space whose underlying set is the disjoint union Z = X ∪ Y , endowed with the topology a basis of which is given by the sets U ∪ V , where

(i) U is open in X, V is open in Y ,

(ii) the closure of φ^{−1}(V) \ U in X is compact (or empty). Note that, by construction, the open sets of the space X are open also in Z and that the intersections of Y with the open sets of Z are exactly the open sets in the original topology of Y . In particular, the inclusions X → Z and Y → Z are homeomorphic embeddings. Furthermore, Z is Hausdorff. To see this, it suffices to consider points x ∈ X, y ∈ Y . Pick a relatively compact open neighborhood U of x in X. Then y lies in the open subset (X \ \overline{U}) ∪ Y of Z and U ∩ ((X \ \overline{U}) ∪ Y) = Ø. Thus Z is Hausdorff.

2.7. Proposition. Suppose that in the above gluing construction 2.6 the space Y is compact. Then for the topological space Z = X⊔_{φ}Y the following assertions hold:

(i) Z is compact.

(ii) In the given topology of Z a net ⟨x_{i}⟩ with x_{i} ∈ X converges to an element y ∈ Y if and only if the following two conditions hold:

(a) lim x_{i} = ∞ in X,

(b) lim φ(x_{i}) = y in Y . (iii) The following assertions are equivalent:

(c) For each non-empty open subset V of Y the set φ^{−1}(V) is not relatively compact in X.

(d) X is dense in Z.

Proof. (i) Consider a covering C = {U_{i} ∪ V_{i}}_{i∈I} of Z by open subsets in the basis defined in 2.6. Since Y is compact, there is a finite set F ⊆ I such that Y = ⋃_{f∈F} V_{f}. For each f ∈ F there is a compact set K_{f} ⊆ X with φ^{−1}(V_{f}) \ U_{f} ⊆ K_{f}. Let K_{F} = K ∪ ⋃_{f∈F} K_{f}. This set is compact, hence there exists a finite set J ⊆ I such that K_{F} ⊆ ⋃_{j∈J} U_{j}. The family {U_{f} ∪ V_{f}}_{f∈F} ∪ {U_{j} ∪ V_{j}}_{j∈J} is finite and covers K_{F} ∪ Y , by construction. Also, the set X \ K_{F} is covered by {U_{f}}_{f∈F} , so we have found a finite subcovering of C. Thus Z is compact.

(ii) Suppose first that lim x_{i} = y in Z. Then for every compact subset C of X the union W = (X \ C) ∪ Y is an open neighborhood of y in Z, hence eventually x_{i} ∈ W ∩ X = X \ C, which shows (a). To prove (b), consider an arbitrary open neighborhood V of y in Y . Since φ^{−1}(V) ∪ V is an open neighborhood of y in Z, we have x_{i} ∈ φ^{−1}(V) ∪ V and hence φ(x_{i}) ∈ V , for all sufficiently large indexes i. Thus lim φ(x_{i}) = y in Y .

Assume now that the conditions (a) and (b) are satisfied. Consider an open set U ∪ V in the basis with y ∈ U ∪ V . Then y ∈ V and condition (b) implies that there exists an index i_{1} ∈ I such that φ(x_{i}) ∈ V whenever i ≥ i_{1} . Thus x_{i} ∈ φ^{−1}(V), for i ≥ i_{1} . On the other hand, we find a compact set C ⊆ X satisfying φ^{−1}(V) \ U ⊆ C. Condition (b) yields the existence of an i_{2} ∈ I such that x_{i} ∈ X \C, for i ≥ i_{2} . Choose an i_{0} ∈ I with i_{0} ≥ i_{1} and i_{0} ≥ i_{2} . If i ≥ i_{0} , then x_{i} belongs to U (because x_{i} ∈ φ^{−1}(V), x_{i} ∉ C, and φ^{−1}(V) \ U ⊆ C). We conclude that lim x_{i} = y in X ⊔_{φ} Y .

Assertion (iii) is an immediate consequence of the definition of the topology of Z.

2.8. Remark. (i) Note that 2.7(ii) says that the closure of X in Z is the union of X with those points y ∈ Y for which there exist arbitrarily small compact neighborhoods K_{y} such that φ^{−1}(K_{y}) is not compact. The restriction of φ to the set φ^{−1}(Y \ cl_{Z}(X)) is a perfect map.

(ii) If K is contained in a compact set K_{1} and φ_{1} is the restriction of φ to X \ K_{1} then X ⊔_{φ} Y = X ⊔_{φ_{1}} Y (as topological spaces).

2.9. Construction. If in the above topological construction the space X is a locally compact topological semigroup acting two-sidedly on a compact topological semigroup Y then X ⊔_{φ} Y becomes a semigroup if we introduce the natural multiplication described in 2.4(i). In order to guarantee that this multiplication is (jointly) continuous we have to assume in addition that X is a perfect semigroup, that the s-action is continuous, and that φ satisfies the following conditions:

(i) φ is asymptotically a homomorphism, that is, if ⟨(s_{i}, t_{i})⟩ is a net in X × X with s_{i} → ∞ and t_{i} → ∞ then

lim φ(s_{i}t_{i}) = (lim φ(s_{i}))(lim φ(t_{i}))

whenever the limits lim φ(s_{i}) and lim φ(t_{i}) exist in Y .

(ii) φ is asymptotically equivariant, that is, if ⟨(s_{i}, t_{i})⟩ is a net in X × X with s_{i} → ∞ then

lim φ(s_{i}t_{i}) = (lim φ(s_{i}))(lim t_{i})
lim φ(t_{i}s_{i}) = (lim t_{i})(lim φ(s_{i}))

whenever lim t_{i} exists in X and lim φ(s_{i}) exists in Y .

If X is dense in Z then by the continuous extension theorem (cf., e.g., [1], p. 81, Theorem 1) these conditions indeed guarantee that Z = X ⊔_{φ} Y becomes a topological semigroup (details are left to the reader). Since no ambiguities are to be feared we also denote this semigroup with X ⊔_{φ} Y .

Obviously, our construction yields a topological semigroup compactification of X. We formally write this compactification as (X ⊔_{φ} Y, i), where i: X → Z denotes the inclusion.

2.10. Remark. (i) In the simplest version of our construction the semigroup Y is singleton and φ is the constant map X → Y . Here our assumptions boil down to the condition that X is perfect, and we get the familiar one point compactification of X with a zero element at infinity.

(ii) If Y is compact but X is not dense in Z = X ⊔_{φ} Y then it is not clear that the assumptions in our construction imply the joint continuity of the multiplication in Z.

(iii) The crux of the construction lies in finding an appropriate mapping φ. The existence of a map φ satisfying 2.9(i),(ii) can be interpreted as an “asymptotic property” of the topological semigroup X.

2.11. Induced compactifications. If X_{1} is a closed subsemigroup of X then the above compactification (X ⊔_{φ} Y, i) of X naturally induces the compactification (\overline{i(X_{1})}, i_{1}) of X_{1} , where i_{1}: X_{1} → \overline{i(X_{1})}, i_{1}(x) = i(x). Set Y_{1} = \overline{i(X_{1})} ∩ Y . If φ^{−1}(\overline{φ(X_{1} \ K)} \ Y_{1}) is relatively compact then the induced compactification can be written as (X_{1} ⊔_{φ_{1}} Y_{1}, i_{1}), where φ_{1}: X_{1} \ K_{1} → Y_{1} , φ_{1}(x) = φ(x), and K_{1} is a compact subset of X_{1} which contains K ∩ X_{1} as well as X_{1} ∩ φ^{−1}(φ(X_{1} \ K) \ Y_{1}). Note that φ^{−1}(\overline{φ(X_{1} \ K)} \ Y_{1}) is empty (in fact, \overline{φ(X_{1} \ K)} \ Y_{1} = Ø) if for every open subset V of Y meeting φ(X_{1} \ K) the set φ^{−1}(V) ∩ X_{1} is not relatively compact .

3. A natural s-action of Sl(2,ℝ)

3.1. Our next goal is to apply the ⊔-construction and Example 2.3(ii) to get compactifications of subsemigroups of Sl(2,ℝ). For this task it is convenient to change to the Lie algebra setting. We also need some preparations from [2].

3.2. Notation and basic facts. (i) Throughout these notes ℝ^{+} will denote the set of strictly positive reals, and ℝ_{0}^{+} the set of nonnegative reals.

(ii) We abbreviate Sl(2,ℝ) to Sl_{2} . Following [11] we write H = \binom{1 0}{0 −1}, P = \binom{0 1}{0 0}, Q = \binom{0 0}{1 0}. Furthermore, we denote the Killing form of 𝔰𝔩(2, ℝ) by Kill and put Kill^{+} = \{X ∈ 𝔰𝔩(2, ℝ) | det(X) = −\frac{1}{8}Kill(X, X) < 0\}. The set

Hyp = {X ∈ 𝔰𝔩(2, ℝ) | − det(X) = 1} = {gHg^{−1} | g ∈ Sl_{2}}

is called the fundamental hyperboloid. For a point X ∈ Hyp let 𝔭_{X} [𝔮_{X}] denote the one-dimensional eigenspace of ad X with eigenvalue 2 [-2]. Then hor(X) = X+𝔭_{X} is called the horizontal line through X, and vert(X) = X+𝔮_{X} is called the vertical line through X. Explicit formulas for the so defined mappings hor and vert can be found in [2], 4.5.

(iii) The set of all horizontal [vertical] lines in Hyp is denoted with Hor [Vert]. We let G_{ℓ} be the parabolic subgroup {1, −1} exp(ℝH + ℝP), and, similarly, G_{r} = {1, −1} exp(ℝH + ℝQ). We identify Hor with the flag manifold Sl_{2}/G_{ℓ}, and Vert with G_{r}\Sl_{2} , so that hor(H) corresponds to G_{ℓ}, and vert(H) to G_{r}. Note that under this convention hor(gHg^{−1}) [vert(gHg^{−1})] is identified with gG_{ℓ} [G_{r}g^{−1} ].

(iv) The map c: Hyp → Hor × Vert, X ↦ (hor(X), vert(X)), is a topological embedding and the rectangular multiplication of Hor × Vert restricts to a partial product ^{♢} on Hyp which we call the diamond product. If hor(X) meets vert(Y), equivalently: if hor(X) ≠ − vert(Y), then hor(X) ∩ vert(Y) = {X^{♢}Y }. For example, it can be checked easily that H^{♢}(αH + βP + γQ) = H + \frac{2β}{1+α}P whenever α ≠ −1.

(v) The restriction of the exponential map to Kill^{+} is a diffeomorphism onto the open set of all matrices in Sl_{2} with trace > 2. We denote with rlog the reduced logarithm exp(Kill^{+}) → Hyp which to every element exp(X), X ∈ Kill^{+} , assigns the normalized vector − det(X)^{−1} ·X. The map rlog will be the essential ingredient in defining the asymptotic homomorphism φ of our construction.

3.3. A partial s-action on Hyp. We now turn to the s-action of 2.3(i) in the special case where G = Sl_{2} and G_{ℓ}, G_{r} are defined as in 3.2(iii) above. This s-action induces a partial s-action ^{∘} of Sl_{2} on Hyp by the rules

g^{∘}X = c^{−1}(gc(X)), and X^{∘}g = c^{−1}(c(X)g),

whenever gc(X) ∈ c(Hyp), c(X)g ∈ c(Hyp), respectively. In terms of the diamond product:

g^{∘}X = (gXg^{−1})^{♢}X and X^{∘}g = X^{♢}(g^{−1}Xg),

whenever the diamond products involved exist. Note that our partial s-action is continuous.

We are interested in asymptotic properties of the action ^{∘}. These involve, however, the application of the function rlog, which implies that we have to restrict our partial s-action to a suitable open domain. Moreover, we need a little lemma enabling us to simplify the necessary calculations.

3.4. Lemma.

(i) If U is a neighborhood of the identity in Sl_{2} then {gHg^{−1} | g ∈ U} is a neighborhood of H in Hyp.

(ii) If ⟨X_{n}⟩ is a sequence in Hyp converging to H then there exist a subsequence ⟨X_{m}⟩ of it and a sequence ⟨g_{m}⟩ in Sl_{2} such that lim g_{m} = 1 and g_{m}X_{m}g_{m}^{−1} = H for every index m.

Proof. (i) By [2], 4.6(ii), we know that the homogeneous space Sl_{2}/Z exp(ℝH) (where Z = {1, −1} is the center of Sl_{2} ) is homeomorphic with Hyp via the map g · Z exp(ℝH) ↦ gHg^{−1} . The assertion follows now from the openness of the canonical quotient map Sl_{2} ↦ Sl_{2}/Z exp(ℝH).

(ii) We choose a compatible metric d on Sl_{2} and write U_{m} = {g ∈ Sl_{2} | d(1, g) < \frac{1}{m}\}. By (i) we find a subsequence ⟨X_{m}⟩ of ⟨X_{n}⟩ such that X_{m} ∈ {gHg^{−1} | g ∈ U_{m}} for every m. Then for suitable g_{m} ∈ U_{m} we have X_{m} = g_{m}Hg_{m}^{−1} and the assertion follows.

3.5. The standard domain. We let

dom_{ℓ} =^{def} {(g, X) ∈ Sl_{2} ×Hyp | ∃T > 0 : g exp(tX) ∈ exp(Kill^{+}) whenever t > T },

dom_{r} =^{def} {(X, g) ∈ Hyp ×Sl_{2} | ∃T > 0 : exp(tX)g ∈ exp(Kill^{+}) whenever t > T },

and define the standard domain of ^{∘} to be the union dom = dom_{ℓ} ∪ dom_{r} . The following assertions hold:

(i) (g, X) ∈ dom_{ℓ} if and only if (−X, g^{−1}) ∈ dom_{r}.

(ii) If (g, X) ∈ dom_{ℓ} then (hgh^{−1}, hXh^{−1}) ∈ dom_{ℓ}, for every h ∈ Sl_{2} .

(iii) (g, H) ∈ dom_{ℓ} if and only if g ∈ O = \{\binom{a b}{c d} ∈ Sl_{2} | a > 0\}.

(iv) If (g, X) ∈ dom_{ℓ} then the diamond product (gXg^{−1})^{♢}X exists. In particular, dom_{ℓ} is a domain of the partial left action defined in 3.3.

(v) All of the three sets dom_{ℓ}, dom_{r} and dom are open.

Proof. Assertion (i) is immediate, (ii) holds by the invariance of exp(Kill^{+}) under inner automorphisms, (iii) follows from 8.8 of [2].

(iv) In view of (ii) we can assume without loss of generality that X = H. Thus g ∈ O, by (iii). Using the formula 4.5 of [2] for hor(gHg^{−1}) we see that hor(gHg^{−1}) ≠ − vert(H), so the diamond product (gHg^{−1})^{♢}H exists.

(v) By (i) and (ii) it suffices to show that dom_{ℓ} is a neighborhood of (g, H) if (g, H) ∈ dom_{ℓ}. For an element (g, H) ∈ dom_{ℓ} let U be a 1neighborhood in Sl_{2} and V a neighborhood of g such that h^{−1}V h ⊆ O for every h ∈ U. Hence V ⊆ ⋂_{h∈U} hOh^{−1} . For arbitrary x ∈ V and h ∈ U there exist y ∈ O and T > 0 such that x = hyh^{−1} and y exp(tH) ∈ exp(Kill^{+}), for every t > T. Hence x exp(thHh^{−1}) = h(y exp(tH))h^{−1} ∈ exp(Kill^{+}), for every t > T. Thus, by 3.4(i), the set V ×{hHh^{−1} | h ∈ U} is a neighborhood of (g, H) contained in dom_{ℓ}.

3.6. Remark. The set dom does not contain all points where our partial s-action is defined. For example, let g = −1. Then the diamond products (gHg^{−1})^{♢}H = H^{♢}H = H and H^{♢}(g^{−1}Hg) = H exist, but neither of (g, H), (H, g) lies in the standard domain.

3.7. Elementary properties of our partial s-action. The following assertions are immediate consequences of the definition of ^{∘}, proofs are left to the reader.

(i) If (g, X) ∈ dom_{ℓ} then g^{∘}X ∈ vert X, if (X, g) ∈ dom_{r} then X^{∘}g ∈ hor X.

(ii) h(g^{∘}X)h^{−1} = (hgh^{−1})^{∘}(hXh^{−1}) for every (g, X) ∈ dom_{ℓ} and every h ∈ Sl_{2} .

(iii) X^{∘}g = −(g^{−1}^{∘}(−X)) for every (X, g) ∈ dom_{r}.

3.8. Asymptotic formulas.

(i) For any (g, X) ∈ dom_{ℓ} we have the equality

g^{∘}X = lim_{{g′→g \atop t→+∞} ; X′→X} rlog(g′ exp(tX′)).

In particular, g^{∘}X = lim_{t→+∞} rlog(g exp(tX)).

(ii) For any (X, g) ∈ dom_{r} the equality

X^{∘}g = lim_{{t→+∞ \atop X′→X} ; g′→g} rlog(exp(tX′)g′)

holds. In particular, X^{∘}g = lim_{t→+∞} rlog(exp(tX)g).

Proof. (i) We first show that

(∗)  g^{∘}X = lim_{{g′→g \atop t→+∞}} rlog(g′ exp(tX)).

For this we may assume (by 3.7(ii)) that X = H. Let g = \binom{a b}{c d}. We have seen in 3.5(iii) that a > 0. Remark 8.8 of [2] implies that

lim_{{g′→g \atop t→+∞}} rlog(g′ exp(tH)) = H + \frac{2c}{a}Q.

On the other hand, a straightforward calculation yields (gHg^{−1})^{♢}H = H + \frac{2c}{a}Q, so (∗) holds.

We prove now the asserted equality assuming again that X = H. Consider the sequences ⟨g_{n}⟩ in Sl_{2} , ⟨t_{n}⟩ in ℝ^{+} , and ⟨X_{n}⟩ in Hyp such that g_{n} → g, t_{n} → +∞, and X_{n} → H. Applying 3.4(ii) there is a subsequence ⟨X_{m}⟩ of ⟨X_{n}⟩ and a sequence ⟨h_{m}⟩ in Sl_{2} converging to 1 such that h_{m}X_{m}h_{m}^{−1} = H, for every index m. Then

rlog(g_{m} exp(t_{m}X_{m})) = h_{m}^{−1} rlog(h_{m}g_{m}h_{m}^{−1} exp(t_{m}H))h_{m}

converges to g^{∘}H by (∗), which finishes the proof.

Assertion (ii) follows from (i) above and 3.7(iii).

3.9. Remark. In the proof of 3.8 we encountered the special formula \binom{a b}{c d}^{∘}H = H + \frac{2c}{a}Q, which is valid for \binom{a b}{c d} ∈ Sl_{2} with a > 0. This formula has other interesting consequences, which we record for later use:

(i) Let (g, X) ∈ dom_{ℓ} [ (X, g) ∈ dom_{r}]. Then g^{∘}X = X [X^{∘}g = X] if and only if g ∈ exp(ℝ hor(X)) [ g ∈ exp(ℝ vert(X)) ].

(ii) Suppose that the diamond products X^{♢}Y and Y ^{♢}X exist. Then

lim_{{t→∞ \atop X′→X}} exp(tX′)^{∘}Y = lim_{{t→∞ \atop Y ′→Y}} X^{∘} exp(tY′) = X^{♢}Y.

If X ≠ Y ^{♢}X [Y ≠ X^{♢}Y] then exp(ℝ^{+}Y)^{∘}X [ Y ^{∘} exp(ℝ^{+}X) ] is the open line segment between X and Y ^{♢}X [ Y and Y ^{♢}X].

(iii) If (\binom{a b}{c d}, H + γQ) ∈ dom_{ℓ} then \binom{a b}{c d}^{∘}(H + γQ) = H + \frac{4c+2dγ}{2a+bγ} Q.

(iv) If (H + βP, \binom{a b}{c d}) ∈ dom_{r} then (H + βP)^{∘} \binom{a b}{c d} = H + \frac{4b+2dβ}{2a+cβ} P.

Proof. (i) follows from the formula \binom{a b}{c d}^{∘}H = H + \frac{2c}{a}Q after conjugating the element X to the element H.

(ii) The equality lim_{{t→∞ \atop X′→X}} exp(tX′)^{∘}Y = X^{♢}Y can be checked by a straight-

forward calculation after conjugating Y to H. In view of 3.7(iii) this equality yields the second one. For the last assertion of (ii) we now have only to observe that t ↦ exp(tY)^{∘}X is injective.

(iii) An easy calculation shows that

\binom{a b}{c d}^{∘}(H + γQ) = e^{γ/2·adQ} (e^{−γ/2·adQ} (\binom{a b}{c d}) ^{∘}H) = H + \frac{4c + 2dγ}{2a + bγ} Q,

and this establishes the assertion. Assertion (iv) follows from (iii) and formula 3.7(iii).

4. Compact Rees extensions of subsemigroups of Sl(2,ℝ)

4.1. In the present section we apply the general construction of Section 2 to produce topological semigroup compactifications of subsemigroups of Sl_{2} . This construction will yield, in particular, explicit descriptions for all injective compactifications for a large class of perfect subsemigroups of Sl_{2} (including the exponential ones).

The following theorems summarize some results about exponential subsemigroups of Sl_{2} and about the diamond product, needed for our constructions below. (Cf. [2] 3.8, 5.5, 7.12, 8.9, 8.10(i), 9.4(i) and [3] 6.13.)

4.2. Theorem. Let W be the Lie wedge of a closed three dimensional exponential subsemigroup Σ of Sl_{2} . Then W is a three dimensional Lie semialgebra in 𝔰𝔩(2, ℝ) and Σ = exp W and the following assertions are equivalent:

(i) Σ is perfect.

(ii) W is contained in {0} ∪ Kill^{+} .

(iii) W ∩ Hyp = rlog(Σ \ {1}) is a compact connected rectangular band semigroup with respect to the diamond product ^{♢}.

4.3. Theorem. Suppose that X, Y ∈ Hyp such that both diamond products X^{♢}Y and Y ^{♢}X exist. Then

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

4.4. Conventions I. Throughout the rest of this section we assume that W is the Lie wedge of a closed perfect exponential subsemigroup Σ of Sl_{2} with dim Σ = 3. We write (slightly at variance with the notation in [2]) D = W ∩Hyp, also, D is considered as a semigroup with respect to ^{♢}. In the terminology of [2] (7.2), D is the closure of a type 0 rectangular domain. By 3.8 the partial s-action ^{∘} restricts to an s-action of Σ on D, which we also denote with ^{∘}.

4.5. Proposition. Let φ: Σ \ {1} → D be defined by φ(s) = rlog(s). Then the following assertions hold:

(i) The map φ is asymptotically a homomorphism and asymptotically equivariant.

(ii) For every X ∈ D the set φ^{−1}(X) = exp(ℝ^{+}X) is not relatively compact in Σ.

Proof. (i) That φ is asymptotically a homomorphism is immediate from 4.3. That φ is asymptotically equivariant follows from 3.8.

Assertion (ii) is obvious.

4.6. Remark. It is also possible (though less convenient) to define φ in terms of a global function φ^{∗}: (\overline{exp(Kill^{+})} \ \{1\}) → (G/G_{ℓ}) × (G_{r}\G), where G = Sl_{2} and G_{ℓ}, G_{r} are as in 3.2(iii). The map φ^{∗} is defined by φ^{∗}(g) = (g_{ℓ}G_{ℓ}, G_{r}g_{r}), where g_{ℓ}, g_{r} are elements with g ∈ g_{ℓ} exp(ℝ_{0}^{+}H + ℝP)g_{ℓ}^{−1} ∩ g_{r}^{−1} exp(ℝ_{0}^{+}H + ℝQ)g_{r}. (Note that the cosets g_{ℓ}G_{ℓ}, G_{r}g_{r} are uniquely defined.) The functions φ and φ^{∗} are related to each other via the map c: Hyp → (G/G_{ℓ}) × (G_{r}\G) of 3.2(iv), by the formula c(φ(s)) = φ^{∗}(s), for all s ∈ Σ \ {1}.

4.7. The semigroup Σ^{D}. By 4.5 Construction 2.9 yields a (topological) semigroup compactification (Σ ⊔_{φ} D, i) of Σ, with inclusion map i. We henceforth abbreviate Σ ⊔_{φ} D to Σ^{D}.

4.8. Remark. (i) Note that Σ^{D} is a nonabelian compact uniquely divisible (UDC) topological semigroup. It is a three dimensional analogue of the so-called “affine triangle”

\{( \begin{matrix}x & y \\ 0 & 1 \end{matrix} ) ∈ M(2, ℝ) | 0 ≤ x, 0 ≤ y, x + y ≤ 1\} .

The semigroup Σ^{D} is a foliated union of copies of the affine triangle.

(ii) In particular, if X ∈ D then \overline{i(exp(ℝ_{0}^{+}X))} = i(exp(ℝ_{0}^{+}X)) ∪ \{X\}. Thus if X ≠ Y ∈ D then \overline{i(exp(ℝ_{0}^{+}X))} ∩ \overline{i(exp(ℝ_{0}^{+}Y))} = \{1\}.

(iii) The minimal ideal D of Σ^{D} is a rectangular band, so all maximal subgroups of Σ^{D} are trivial.

(iv) Let ρ be a finite dimensional representation of Σ^{D}, such that ρ(1) is the identity. By Theorem 2.1 of [4] any congruence on Σ is either a Rees congruence or the identity, so det ∘ρ(Σ) = {1}. This means that the matrices in ρ(Σ) have determinant 1, hence are invertible. It follows that ρ(Σ^{D}) is a compact topological group, which implies ρ(Σ^{D}) = {1}, since every continuous homomorphism of Σ into a compact topological group is trivial (see Proposition 6.2 below). Thus all finite dimensional representations of Σ^{D} are trivial.

(v) The so-called ABC-theorem of Brown and Friedberg [5] states, among other things, that if S satisfies all of the following conditions (a)–(e) then S has a faithful real n-dimensional representation:

(a) E(S) = {1} ∪ M(S).

(b) M(S) is homeomorphic to an (n − 1)-cell.

(c) The centralizer of each idempotent e ∈ M(S) is isomorphic with the (multiplicative) semigroup [0, 1].

(d) S \ M(S) admits left cancellation.

(e) M(S) is a left zero semigroup.

Since conditions (a)–(d) are obviously satisfied for S = Σ^{D} and n = 3 the above observation (iv) shows that the ABC-theorem is no longer valid if condition (e) is omitted.

We next pass to a construction where the H-classes are nontrivial.

4.9. Conventions II. To avoid tedious calculations we henceforth also assume that D = W ∩ Hyp contains the element H. This can always be enforced by applying a suitable inner automorphism, so we do not lose generality. We write

h := hor(H) ∩ D, v := vert(H) ∩ D.

Note that D = v^{♢}h.

4.10. The paragroup M(D, K). Let (K, k) be a topological group compactification of the additive group of real numbers. Then we turn the compact space M(D, K) = v × K × h into a paragroup, with multiplication

(X, g, Y)(X′, g′, Y′) = (X, gσ(Y, X′)g′, Y′),

where, for any Y = H + βP ∈ h, X′ = H + γQ ∈ v we let

σ(H + βP, H + γQ) = k(log(1 + \frac{βγ}{4} )).

To justify the definition of the sandwich map σ, note first that for Y, X′ as above we always have 4 + βγ ≠ 0. This follows from [2] 4.15 and the fact that the diamond product X′^{♢}Y exists. By 4.2 the set D is connected, so we conclude that 4 + βγ has the same sign for all pairs (H + βP, H + γQ) ∈ h × v, this sign must be positive since (H, H) ∈ h × v.

4.11. The s-action of Σ on M(D, K). With the aid of the s-action ^{∘} of Σ on D we next define an s-action of Σ on M(D, K): For s = \binom{a b}{c d} ∈ S and (H + γQ, g, H + βP) ∈ v × K × h we put

\begin{aligned}s(H + γQ, g, H + βP) & = (s^{∘}(H + γQ), k(log(a + \frac{bγ}{2} ))g, H + βP) \\ (H + γQ, g, H + βP)s & = (H + γQ, gk(log(a + \frac{cβ}{2} )), (H + βP)^{∘}s). \end{aligned}

Note that the condition s exp(t(H + γQ)) ∈ exp(Kill^{+}) [exp(t(H + βP))s ∈ exp(Kill^{+})] for every t ∈ ℝ^{+} implies that a + \frac{bγ}{2} > 0 [a + \frac{cβ}{2} > 0]. It is readily verified by direct calculation that our definition indeed yields an s-action of Σ on M(D, K).

4.12. Example. Suppose that s = exp(tX), where X = (H + γQ)^{♢}(H + βP) and t > 0. Then we get

s(H + γQ, g, H + βP) = (H + γQ, k(t)g, H + βP).

A similar formula holds for (H + γQ, g, H + βP)s.

For the next proposition we recall that by 3.5(iii) we have a > 0 for every \binom{a b}{c d} ∈ Σ.

4.13. Definition and Proposition. Let M(D, K) be the paragroup of 4.10. Then we define a continuous map Φ: Σ \ {1} → M(D, K),

s = \binom{a b}{c d} ↦ Φ(s) = (φ(s)^{♢}H, k(log(a)), H^{♢}φ(s)).

(Since the choice of D, K will always be clear from the context we shall omit any reference to D, K in the notation for Φ.)

The following assertions hold:

(i) The map Φ is asymptotically a homomorphism and asymptotically equivariant.

(ii) For every non-empty open subset V of M(D, K) the set Φ^{−1}(V) is not relatively compact in Σ.

Proof. (i) To verify that Φ is asymptotically a homomorphism choose nets ⟨s_{i} = \binom{a_{i} b_{i}}{c_{i} d_{i}}⟩ and ⟨s′_{i} = \binom{a′_{i} b′_{i}}{c′_{i} d′_{i}}⟩ in Σ such that s_{i} → ∞, s′_{i} → ∞, Φ(s_{i}) → (H +γQ, g, H +βP), and Φ(s′_{i}) → (H +γ′Q, g′, H +β′P). We have to show that the net ⟨Φ(s_{i}s′_{i})⟩ converges to (H + γQ, gk(log(1 + \frac{βγ′}{4} ))g′, H + β′P). By 4.5(i) the map φ is asymptotically a homomorphism, so φ(s_{i}) → (H + γQ)^{♢}(H + βP) and φ(s′_{i}) → (H + γ′Q)^{♢}(H + β′P) imply that φ(s_{i}s′_{i}) → (H + γQ)^{♢}(H + β′P). Thus it remains to verify that

(∗)  k(log(a_{i}a′_{i} + b_{i}c′_{i})) → gk(log(1 + \frac{βγ′}{4} ))g′.

To accomplish this, observe first that the equality a_{i}a′_{i} + b_{i}c′_{i} = a_{i} (1 + \frac{b_{i}}{a_{i}} \frac{c′_{i}}{a′_{i}} ) a′_{i} implies that

k(log(a_{i}a′_{i} + b_{i}c′_{i})) = k(log(a_{i}))k (log (1 + \frac{b_{i}}{a_{i}} \frac{c′_{i}}{a′_{i}} )) k(log(a′_{i})).

Since rlog(s_{i}) = φ(s_{i}) → (H + γQ)^{♢}(H + βP) = \frac{1}{4+βγ} ((4 − βγ)H + 4βP + 4γQ) we obtain (using the formula for the exponential function) that

lim \frac{b_{i}}{a_{i}} = \frac{\frac{4β}{4+βγ}}{1 + \frac{4−βγ}{4+βγ}} = \frac{β}{2} .

Similarly, rlog(s′_{i}) = φ(s′_{i}) → (H + γ′Q)^{♢}(H + β′P) implies that lim \frac{c′_{i}}{a′_{i}} = \frac{γ′}{2} . So, (∗) holds, hence Φ is asymptotically a homomorphism.

We now show that Φ is asymptotically equivariant. Pick nets ⟨s_{i} = \binom{a_{i} b_{i}}{c_{i} d_{i}}⟩ and ⟨s′_{i} = \binom{a′_{i} b′_{i}}{c′_{i} d′_{i}}⟩ in Σ such that s_{i} → s = \binom{a b}{c d} ∈ S, s′_{i} → ∞, and Φ(s′_{i}) → (H +γ′Q, g, H +β′P). We show that Φ(s_{i}s′_{i}) → s(H +γ′Q, g, H +β′P).

The convergence Φ(s′_{i}) → (H + γ′Q, g, H + β′P) implies that φ(s′_{i}) → (H + γ′Q)^{♢}(H + β′P), and, by 4.5(i), that φ(s_{i}s′_{i}) → s^{∘}(H + γ′Q)^{♢}(H + β′P). Thus it remains to prove that

(∗∗)  k(log(a_{i}a′_{i} + b_{i}c′_{i})) → k (log (a + \frac{bγ′}{2} )) g.

The equality a_{i}a′_{i} + b_{i}c′_{i} = (a_{i} + b_{i} \frac{c′_{i}}{a′_{i}})a′_{i} implies that

k(log(a_{i}a′_{i} + b_{i}c′_{i})) = k (log (a_{i} + b_{i} \frac{c′_{i}}{a′_{i}} )) k(log(a′_{i})).

We have seen above that lim \frac{c′_{i}}{a′_{i}} = \frac{γ′}{2} , so we obtain (∗∗), and thence Φ(s_{i}s′_{i}) → s(H+γ′Q, g, H+β′P). By the same arguments Φ(s′_{i}s_{i}) → (H+γ′Q, g, H+β′P)s, thus Φ is asymptotically equivariant.

(ii) Let V be a non-empty open subset of M(D, K). Consider an arbitrary element (Y, g, Z) ∈ V and an open subset U of K containing g such that

{Y } × U × {Z} ⊆ V.

Let X = Y ^{♢}Z = αH + βP + γQ. We know from 8.2 of [2] that 1 + α > 0. The properties of the compactifications of the additive group ℝ imply that there exists a sequence ⟨r_{n}⟩ of positive reals converging to +∞ such that k(r_{n}) ∈ U for every n ∈ ℕ. Since 1 + α > 0 we find for every n ∈ ℕ a positive real t_{n} such that cosh(t_{n}) + α sinh(t_{n}) = e^{r_{n}}. Then t_{n} → ∞. Since Φ(exp(t_{n}X)) = (Y, k(r_{n}), Z) ∈ V for every n ∈ ℕ we conclude that Φ^{−1}(V) is not relatively compact in Σ.

4.14. The semigroup Σ^{D,K}. By 4.13 Construction 2.9 yields a (topological) semigroup compactification (Σ ⊔_{Φ} M(D, K), j) of Σ with inclusion map j. We henceforth abbreviate Σ ⊔_{Φ} M(D, K) to Σ^{D,K}.

4.15. Connections between Σ^{D} and Σ^{D,K}. The map Σ^{D,K} → Σ^{D} defined by

\begin{matrix}a & ↦ a for a ∈ S \\ (X, g, Y) & ↦ X^{♢}Y for (X, g, Y) ∈ M(D, K) \end{matrix}

is a continuous and surjective homomorphism whose kernel congruence is the H-relation on M(D, K).

4.16. Induced compactifications of a subsemigroup of Σ. If S is a closed subsemigroup of Σ then our compactifications (Σ^{D}, i) and (Σ^{D,K}, j) naturally induce compactifications of S, namely (\overline{i(S)}, i_{S}) and (\overline{j(S)}, j_{S}), where i_{S}(x) = i(x) and j_{S}(x) = j(x) for all x ∈ S.

4.17. Proposition. Let S be a closed subsemigroup of our exponential semigroup Σ. We write φ′ for the restriction of φ to S \\{1\}, and D′ = \overline{φ′(S \ \{1\})}. Then \overline{i(S)} = i(S) ∪ D′ and the compact semigroup \overline{i(S)} can be written as the ⊔-product S ⊔_{φ′} D′.

Proof. Since rlog(s^{n}) = rlog(s), for every s ∈ S \ {1}, n ∈ ℕ, and s^{n} → ∞ this assertion follows by a straightforward application of the definition of the ⊔-product (cf. 2.11).

4.18. For the next proposition we recall from [2] 10.1, that the umbrella set Umb(A) of a subset A in a Lie group G is defined to be the set of all elements X in the Lie algebra of G such that exp(tX) ∈ A for all suitably large positive reals t, say, t ≥ T. (Cf. also 5.2 below.)

4.19. Proposition. We retain the assumptions and the terminology of 4.17, and suppose, in addition, that S has dense and connected interior and that \overline{Umb}(S) = ℝ_{0}^{+}D. Then

(i) D′ = D and \overline{j(S)} = j(S) ∪ M(D, K);

(ii) \overline{Φ(S \ \{1\})} = M(D, K), and the compact semigroup \overline{j(S)} can be written as the ⊔-product S ⊔_{Φ′} M(D, K), where Φ′ denotes the restriction of Φ to S \ {1}.

Proof. (i) The equality D′ = D follows from the fact (cf. [2], 10.5) that the algebraic interior of \overline{Umb}(S) coincides with Umb(Int S).

If X is an interior point of D (in Hyp) then exp([T, ∞[X) ⊆ Int S for some T > 0, thus the proof of 4.13(ii) shows that the closure of j(exp([T, ∞[X)) contains the set \{X^{♢}H\} × K × \{H^{♢}X\}. This implies that M(D, K) ⊆ \overline{j(S)} and the assertion follows.

(ii) The proof of (i) yields the inclusion M(D, K) ⊆ \overline{Φ(S \ \{1\})}. Assertion (ii) follows now from 2.11.

5. Compactifications of subsemigroups of Sl(2,ℝ): Basic Facts

5.1. Notation. (i) Unless stated explicitly otherwise, throughout this section we let S be a closed proper (i.e., S ≠ Sl_{2} ) subsemigroup of Sl_{2} with dense and connected interior. Note that the interior of a connected subsemigroup of Sl_{2} is always connected if it clusters at the identity (cf., e.g., [2], 3.2).

(ii) We use the standard notation for semigroups. In particular, — E( ) stands for the set of idempotents,

— M( ) for the minimal ideal (if it exists),

— H( ) for H-classes.

(iii) We consider a (topological semigroup) compactification of S, for which we write (S^{κ}, κ), or S^{κ} for short. The universal such compactification, the Bohr compactification of S, is denoted by (S^{b}, b_{S}), or S^{b}.

(iv) Since no ambiguities are to be feared we denote both the left and the right natural action of S on its compactification S^{κ} by the dot “.”. Thus s.x = κ(s)x and x.s = xκ(s) for every x ∈ S^{κ} and s ∈ S.

(v) If 0 ≠ X ∈ 𝔰𝔩(2, ℝ) with exp(ℝ^{+}X) ∩ S ≠ Ø then e_{X} denotes the (unique) idempotent in the minimal ideal of the compact abelian semigroup \overline{κ(exp(ℝ^{+}X) ∩ S)}. Since there is no danger of confusion we usually do not specify the particular compactification.

5.2. Frequently used facts about umbrella sets. (Cf. [2] 10.3, 3.4, 10.5)

(i) If A is a subsemigroup of G and exp X lies in the interior of A then X ∈ Umb(A).

(ii) If G = Sl_{2} and A is an open and connected proper subsemigroup of G

then (a) A ⊂ exp(Kill^{+}),

(b) Umb(A) = ℝ^{+} rlog(A) and Umb(A) ⊆ Umb(\overline{A}) ⊆ \overline{Umb}(A),

(c) Umb(A) is the interior of an exponential Lie wedge, i.e., a wedge in \overline{Kill^{+}} which is the intersection of at most four half spaces, each bounded by a Borel algebra.

(d) If X ∈ \overline{Umb}(A) ∩ Hyp and a ∈ \overline{A} then the pairs (X, a) and (a, X) lie in the standard domain and \{X^{∘}a, a^{∘}X\} ⊂ \overline{Umb}(A).

5.3. Proposition. Let x = exp(X) be an interior point of S and suppose that ⟨n_{i}⟩ is a net of positive real numbers with lim n_{i} = +∞ and such that the limit ℓ = lim κ(exp(n_{i}X)) exists. Then

(i) ℓS^{κ}ℓ = M_{X}, where M_{X} is the minimal ideal of the compact abelian semigroup \overline{κ(exp(ℝ_{0}^{+}X) ∩ S)};

(ii) ℓ lies in the minimal ideal M(S^{κ}) of S^{κ}, and M_{X} = H(e_{X});

(iii) there is a unique continuous homomorphism k: ℝ → H(e_{X}) with k(t) = e_{X}. exp(tX) whenever exp(tX) ∈ S, under this homomorphism the image of S ∩ exp(ℝX) is dense in H(e_{X});

(iv) if y = exp Y ∈ S and X, Y lie in the span of a single horizontal [vertical] line then e_{X}e_{Y} = e_{Y} [ e_{X}e_{Y} = e_{X} ], that is, e_{X} and e_{Y} lie in the same minimal right [ left ] ideal.

Proof. Let S_{X} := exp(ℝ_{0}^{+}X)∩S. We observe first that if ⟨t_{i}⟩ is a net of positive real numbers such that lim t_{i} = +∞ and the limit m = lim κ(exp(t_{i}X)) exists, then m ∈ M_{X}. To see this, consider an arbitrary element exp(aX) ∈ S_{X}. The equality exp(t_{i}X) = exp(aX) exp((t_{i} − a)X) holds in S_{X} for sufficiently large t_{i}. We conclude that m ∈ exp(aX).\overline{κ(S_{X})}. This implies that m ∈ M_{X}.

(i) We know that Int S ⊆ exp(Kill^{+}), so there exists an inner automorphism of 𝔰𝔩(2, ℝ) taking rlog(x) to H. Thus it suffices to show the assertion under the assumption that X = tH, where t > 0. Pick s = \binom{a b}{c d} ∈ S. Then a > 0, by 3.5(iii), and we find

exp(n_{i}X)s exp(n_{i}X) = ( \begin{matrix}e^{tn_{i}} & 0 \\ 0 & e^{−tn_{i}} \end{matrix} ) ( \begin{matrix}a & b \\ c & d \end{matrix} ) ( \begin{matrix}e^{tn_{i}} & 0 \\ 0 & e^{−tn_{i}} \end{matrix} )

= ( \begin{matrix}e^{2tn_{i}}a & b \\ c & de^{−2tn_{i}} \end{matrix} )

= ( \begin{matrix}1 & 0 \\ \frac{c}{a}e^{−2tn_{i}} & 1 \end{matrix} ) ( \begin{matrix}ae^{2tn_{i}} & 0 \\ 0 & \frac{e^{−2tn_{i}}}{a} \end{matrix} ) ( \begin{matrix}1 & \frac{b}{a}e^{−2tn_{i}} \\ 0 & 1 \end{matrix} ) .

Since x ∈ Int S we conclude that for n_{i} sufficiently large the elements

u_{i} = ( \begin{matrix}1 & 0 \\ \frac{c}{a}e^{−2tn_{i}} & 1 \end{matrix} ) x and v_{i} = x ( \begin{matrix}1 & \frac{b}{a}e^{−2tn_{i}} \\ 0 & 1 \end{matrix} )

lie in Int S as well. Note that lim u_{i} = lim v_{i} = x.

Now we pass to a suitable subnet such that the elements

y_{i} =^{def} ( \begin{matrix}ae^{2n_{i}−2t} & 0 \\ 0 & \frac{e^{−2n_{i}+2t}}{a} \end{matrix} )

lie in S and such that y = lim κ(y_{i}) exists. Then y ∈ M_{X} and therefore

ℓ.s.ℓ = lim κ(exp(n_{i}X)s exp(n_{i}X)) = lim u_{i}.κ(y_{i}).v_{i} = x.y.x ∈ M_{X}.

Thus we have shown that ℓ.S.ℓ ⊆ M_{X}. We conclude that

M_{X} = ℓM_{X}ℓ ⊆ ℓS^{κ}ℓ = \overline{ℓ.S.ℓ} ⊆ M_{X}.

Hence ℓS^{κ}ℓ = M_{X}.

(ii) By (i) e_{X}S^{κ}e_{X} = ℓS^{κ}ℓ = M_{X} is a group. We know from the general theory of compact semigroups that an idempotent e of a compact topological semigroup C lies in the minimal ideal of C if and only if eCe is a group. Thus e_{X} ∈ M(S^{κ}) and ℓ ∈ M_{X} = e_{X}S^{κ}e_{X} = H(e_{X}) ⊆ M(S^{κ}).

(iii) By (ii) we have a continuous homomorphism k_{0}: [t_{0}, ∞[→ H(e_{X}), t ↦ e_{X}. exp(tX), where t_{0} is a positive real such that exp([t_{0}, ∞[·X) ⊂ S. Define

k(t) = \{ \begin{matrix}e_{X}. exp(tX) & if t ≥ t_{0}, \\ exp(2t_{0}X).(e_{X}. exp(2t_{0} − t)X)^{−1} & if t < t_{0}. \end{matrix}

Then k is a homomorphism and its restriction to ]t_{0}, ∞[ is continuous, so it is continuous everywhere. It is easily checked that k(t) = e_{X}. exp(tX) whenever exp(tX) ∈ S.

(iv) As in the proof of (i) we assume that X = tH. We show the assertion for X, Y lying in the span of the same horizontal line, the other case follows by taking transposes. Then Y = t′(H +βP) for suitable reals t′, β. Replacing y by y^{n} for some n ∈ ℕ we enforce that x′: = exp t′H lies in the interior of S. Then

yx^{n_{i}} = x^{n_{i}} exp(t′(H + βe^{−2n_{i}t}P))

and, since x′ is an inner point of S, for large indexes i the point exp(t′(H + βe^{−2n_{i}t}P) lies in S. Applying κ and passing to limits we thus find that y.ℓ = ℓ.x′ ∈ H(ℓ) and the assertion follows.

5.4. Lemma. Let ⟨x_{i}⟩ be a net in Int S with lim x_{i} = ∞ and such that the limits ℓ = lim κ(x_{i}) and X = lim rlog(x_{i}) exist. Suppose that X ∈ Umb(Int S). Then the following assertions hold:

(i) ℓ ∈ e_{X}S^{κ}e_{X}.

(ii) ℓ lies in the minimal ideal M(S^{κ}) of S^{κ}.

(iii) ℓ ∈ H(e_{X}).

Proof. (i) Since X ∈ Umb(Int S) we find, by virtue of 10.3(ii) in [2], a neighborhood U of X in 𝔰𝔩(2, ℝ) and a positive number t_{0} such that for all Y ∈ U and all t ≥ t_{0} the exponential image exp(tY) is contained in Int S. For convenience, we write X_{i} = rlog(x_{i}). Then x_{i} = exp(t_{i}X_{i}) for some t_{i} ∈ ℝ^{+} . Since lim x_{i} = ∞, we have that lim t_{i} = +∞. Fix some number t > t_{0} . Then for sufficiently large indexes i we have X_{i} ∈ U and t_{i} − t ≥ t_{0} , so the elements exp(tX_{i}) and exp((t_{i} − t)X_{i}) lie in S. Passing to a suitable subnet we enforce that y = lim κ(exp((t_{i} − t)X_{i})) exists. Then

exp(tX).y = lim κ(exp(tX_{i}))κ(exp((t_{i} − t)X_{i})) = lim κ(exp(t_{i}X_{i})) = ℓ.

Similarly y. exp(tX) = ℓ. Thus ℓ ∈ exp(tX).S^{κ} ∩ S^{κ}. exp(tX) for all t > t_{0} . Plugging in a net ⟨t_{j}⟩ with κ(exp(t_{j}X)) → e_{X} and passing to limits we therefore see that ℓ ∈ e_{X}S^{κ}e_{X}, as asserted.

Assertions (ii) and (iii) follow from (i) and Proposition 5.3.

5.5. Remark. For the next lemma note that for every net ⟨X_{i}⟩ in Hyp with lim X_{i} = ∞ there exist a subnet ⟨X_{j}⟩ and positive reals λ_{j} such that ⟨λ_{j}X_{j}⟩ converges to a nonzero nilpotent matrix.

We also recall that Sl_{2}^{+} denotes the set of all matrices in Sl_{2} with nonnegative entries. It is known from [2] 10.5 that every proper subsemigroup of Sl_{2} with dense and connected interior is contained in an exponential subsemigroup, hence, by [2] 3.8, is conjugate to a subsemigroup of Sl_{2}^{+} .

5.6. Lemma. Suppose that S is contained in Sl_{2}^{+} and let ⟨x_{i}⟩ be a net in Int S such that x_{i} → ∞ and rlog(x_{i}) → ∞ in \overline{Umb}(S). We assume that lim λ_{i} rlog(x_{i}) = P for a suitable net ⟨λ_{i}⟩ of positive reals. Then for every x = \binom{v w}{y z} ∈ Int S

(i) lim rlog(xx_{i}) = −H + 2\frac{v}{y} P,
(ii) lim rlog(x_{i}x) = H + 2\frac{z}{y} P.

Proof. Put rlog(x_{i}) = α_{i}H + β_{i}P + γ_{i}Q. Then β_{i}, γ_{i} are nonnegative and thus the relation α_{i}^{2} + β_{i}γ_{i} = 1 implies that γ_{i} ≤ 1/β_{i}. Since λ_{i}β_{i} → 1 we must have β_{i} → ∞ and γ_{i} → 0.

Write x_{i} = \binom{a_{i} b_{i}}{c_{i} d_{i}} = exp(t_{i} rlog(x_{i})) for some positive t_{i}. We claim that b_{i} → +∞. If this is not the case, then ⟨b_{i}⟩ has a convergent subnet ⟨b_{ℓ}⟩. The formula for the exponential function (see, for example, 2.5 of [2]) implies then that t_{ℓ} → 0, hence ⟨a_{ℓ}⟩, ⟨c_{ℓ}⟩, and ⟨d_{ℓ}⟩ are bounded, which contradicts the assumption that x_{i} → ∞. Thus b_{i} → +∞.

The formula for rlog (see 2.5 of [2]) says that for g = \binom{a b}{c d}

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

Thus β_{i} = \frac{2b_{i}}{\sqrt{(a_{i}+d_{i})^{2}−4}} . Since β_{i} → +∞ and b_{i} → +∞ we therefore get \frac{a_{i}+d_{i}}{b_{i}} → 0, hence

\frac{a_{i}}{b_{i}} → 0 and \frac{d_{i}}{b_{i}} → 0. Also \frac{c_{i}}{b_{i}} = \frac{γ_{i}}{β_{i}} → 0.

Now rlog(xx_{i}) = ( \begin{matrix}α_{i}^{∗} & β_{i}^{∗} \\ γ_{i}^{∗} & −α_{i}^{∗} \end{matrix} ) with

\begin{aligned}α_{i}^{∗} & = \frac{va_{i} + wc_{i} − (yb_{i} + zd_{i})}{\sqrt{(va_{i} + wc_{i} + yb_{i} + zd_{i})^{2} − 4}} ⟶ −1 \\ β_{i}^{∗} & = 2\frac{vb_{i} + wd_{i}}{\sqrt{(va_{i} + wc_{i} + yb_{i} + zd_{i})^{2} − 4}} ⟶ 2\frac{v}{y} \\ γ_{i}^{∗} & = 2\frac{ya_{i} + zc_{i}}{\sqrt{(va_{i} + wc_{i} + yb_{i} + zd_{i})^{2} − 4}} ⟶ 0. \end{aligned}

This establishes (i). The proof of (ii) is left to the reader.

5.7. Notation. In the following we use the abbreviation (slightly at variance with the notation in [2]) D := \overline{Umb}(S) ∩ Hyp = \overline{rlog(S ∩ exp(Kill^{+}))}. The set D is called the set of regular directions in S.

5.8. Theorem. Let S be a closed proper subsemigroup of Sl_{2} with dense and connected interior. Let ⟨x_{n}⟩ be a net in Int S with lim x_{n} = ∞ and such that the limit ℓ = lim κ(x_{n}) exists. Then there exists a subnet ⟨x_{i}⟩ of ⟨x_{n}⟩ such that one of the following assertions holds:

(i) X = lim rlog(x_{i}) exists and lies in the interior of D in Hyp. Then ℓ ∈ H(e_{X}) ⊆ M(S^{κ}), where e_{X} is the unique minimal idempotent in the closure of κ(exp(ℝ^{+}X) ∩ S).

(ii) X = lim rlog(x_{i}) exists and lies in the interior of one of the horizontal line segments belonging to the boundary of D in Hyp. Then S^{κ}ℓ ⊆ M(S^{κ}).

(iii) X = lim rlog(x_{i}) exists and lies in the interior of one of the vertical line segments belonging to the boundary of D in Hyp. Then ℓS^{κ} ⊆ M(S^{κ}).

(iv) Either ⟨rlog(x_{i})⟩ converges to a corner point of D or rlog(x_{i}) → ∞. Then S^{κ}ℓS^{κ} = M(S^{κ}).

Proof. We first pass to a subnet ⟨x_{i}⟩ such that either X = lim rlog(x_{i}) exists or rlog(x_{i}) → ∞.

(i) This follows from Lemma 5.4 and the fact that Int D = Umb(Int S) ∩ Hyp (cf. [2] 10.5(ii)).

(ii) We conclude from 3.9(i),(ii) that s^{∘}X lies in the interior of D whenever s lies in the dense subset Int S \ exp(ℝ hor(X)) of S. For such an element s we have lim sx_{i} = ∞ and, by 3.8(i), rlog(sx_{i}) = s^{∘}X. Applying (i) we see that (Int S \ exp(ℝ hor(X))).ℓ ⊆ M(S^{κ}). Since M(S^{κ}) is closed and κ(Int S \ exp(ℝ hor(X))) is dense in S^{κ} this implies the assertion.

(iii) follows from (ii) by applying the anti-automorphism x ↦ x^{−1} . (iv) By minimality it obviously suffices to show S^{κ}ℓS^{κ} ⊆ M(S^{κ}). Assume first that X = lim rlog(x_{i}) is a corner point of D. Then by

3.9(i),(ii) the point s^{∘}X lies in the interior of one of the vertical line segments belonging to the boundary of D in Hyp, provided s ∈ Int S \ exp(ℝ hor(X)). Thus, by (iii) we see that (Int S \ exp(ℝ hor(X))).ℓS^{κ} ⊆ M(S^{κ}), and, consequently, S^{κ}ℓS^{κ} ⊆ M(S^{κ}).

Assume now that rlog(x_{i}) → ∞. Then, not losing generality, we also assume that S ⊆ Sl_{2}^{+} and that we can find positive numbers λ_{i} such that λ_{i} rlog(x_{i}) → P. By Lemma 5.6(i) we know that for every s lying in a dense subset of S the limit lim rlog(sx_{i}) lies in the interior of the line segment vert(−H) ∩ D. For all such points s we infer from (iii) that s.ℓS^{κ} ⊆ M(S^{κ}), so the assertion follows from the compactness of M(S^{κ}).

5.9. Large elements. In the following an element s ∈ S^{κ} is said to be S -large, or large for short, if it is the limit s = lim κ(x_{n}) where ⟨x_{n}⟩ is a net in S with lim x_{n} = ∞. Note that the set of large elements is the intersection ⋂ \overline{κ(A)}, where each A is a cocompact set in S. Since, by our general assumption, S is a subsemigroup of a group it follows without difficulty that the S -large elements form a closed ideal in S^{κ} which contains the growth S^{κ} \ κ(S) of the compactification S^{κ}.

5.10. Theorem. Let S be a closed proper subsemigroup of Sl_{2} with dense and connected interior. Then an S -large element m ∈ S^{κ} lies in the minimal ideal M(S^{κ}) if and only if m ∈ S^{κ}mS^{κ}. In particular, if S contains the identity then every S -large element m belongs to M(S^{κ}).

Proof. By 5.8 we know that, in any case, S^{κ}mS^{κ} ⊆ M(S^{κ}). This establishes our assertion.

5.11. Example. If in the above theorem S is not a monoid then the growth of a compactification of S need not be contained in the minimal ideal.

(i) We first remark that for any topological semigroup with constant multiplication xy = z_{0} the Bohr compactification is equivalent with the Stone-Čech compactification, with constant multiplication x′y′ = β(z_{0}). In this case the minimal ideal of the Bohr compactification never meets the growth of the Bohr compactification, and the growth is nonvoid whenever S is completely regular but noncompact.

(ii) If S is any topological semigroup on a locally compact normal space such that the Rees quotient S/\overline{S^{2}} is not compact then S has a topological semigroup compactification whose growth is nonvoid and does not meet the minimal ideal. This follows from (i) and the fact that S_{1} = S/\overline{S^{2}} is a semigroup with constant multiplication. (Since S is defined on a normal space the Rees quotient S/\overline{S^{2}} is completely regular, hence its Stone-Čech compactification is an embedding, with nonvoid growth since S/\overline{S^{2}} is noncompact.) By the universality of the Bohr compactification this implies that the growth of S^{b} cannot be contained entirely in its minimal ideal.

(iii) Let S be the semigroup consisting of the matrices in Sl_{2}^{+} where both diagonal entries are ≥ 2. This semigroup is connected and has dense interior, since it can be written also as the set of all matrices of the form

s(a, b, u) = ( \begin{matrix}a & b \\ (u − 1)/b & u/a \end{matrix} ) , u ≥ 2a ≥ 4, b > 0.

Note that s(a, b, u) ∈ S^{2} always implies a ≥ 4, thus the Sl_{2} -closed set C = \{s(2, b, 4) | b > 0\} is properly contained in S \ \overline{S^{2}} , so C is not relatively compact in S. Thus by (ii) the growth of S in its Bohr compactification is not contained in the minimal ideal. In fact, (b_{S} | C, \overline{b_{S}(C)}) is equivalent with the Stone-Čech compactification of the reals and \overline{b_{S}(C)} does not meet the minimal ideal.

6. Injectivity and Noninjectivity of Compactifications

6.1. Theorem. Suppose that (S^{κ}, κ) is a topological semigroup compactification of a closed connected proper submonoid S of Sl_{2} with dense interior, and write

I =^{def} κ^{−1}(M(S^{κ})).

(i) An element s ∈ S lies in I if and only if there exists an s′ ≠ s with κ(s) = κ(s′). If I is nonempty then κ(I) = {z}, where z is the zero element of S^{κ}.

(ii) The restriction of κ to S \ I is a homeomorphic embedding.

(iii) The following assertions are equivalent: (a) κ is injective.

(b) κ is a homeomorphic embedding.

(c) I is empty.

(d) κ is not surjective.

Proof. (i) It is shown in [4] 2.1 that every closed congruence on a closed connected submonoid with dense interior of Sl_{2} is either the identity or a Rees congruence. Thus assertion (i) holds whenever the kernel congruence of κ is not the identity, that is, if κ is not injective.

Suppose now that κ is injective and assume that x ∈ I. If x = 1 then S^{κ} = M(S^{κ}) and M(S^{κ}) contains a central idempotent, so S^{κ} is a group. Now the following general proposition, recorded also for later use, shows that S^{κ} is singleton, in contradiction to the injectivity of κ.

6.2. Proposition. Let S be a connected subsemigroup with dense interior of a connected noncompact simple Lie group G. Then every continuous homomorphism of S into a compact topological group is trivial.

Proof. By [11] VII.3.28 the free topological group on S is a covering group of G. Since every connected noncompact simple Lie group is minimally almost periodic this implies that the free topological group on S has no nontrivial continuous homomorphisms into a compact topological group. By the definition of the free topological group on S this finishes the proof.

6.3. Proof of 6.1(i) (ctd). Next we suppose that κ(x) ∈ M(S^{κ}) with x ≠ 1. Then the centralizer of x is nowhere dense in Sl_{2} and we can find a point s ∈ S with x^{2}sx ≠ xsx^{2} . But κ(x).S.κ(x) ⊆ H(κ(x)), hence, since the maximal subgroups of S^{κ} are abelian, κ(x^{2}sx) = κ(x)κ(xsx) = κ(xsx)κ(x) = κ(xsx^{2}). Since this contradicts the injectivity of κ we conclude that I must be empty whenever κ is injective and the assertion follows.

(ii) Suppose that ⟨s_{n}⟩ is a net in S \ I such that lim κ(s_{n}) = κ(s), for some s ∈ S \ I.

If ⟨s_{n′}⟩ were a subnet of ⟨s_{n}⟩ with lim s_{n′} = ∞ then, by Theorem 5.10, κ(s) ∈ M(S^{κ}), a contradiction to s ∈ S \ I.

By (i) every convergent subnet of ⟨s_{n}⟩ must converge to s, so we conclude that lim s_{n} = s and the proof is finished.

(iii) The proof of (i) also shows the implication (a) ⟹ (c). The implication (c) ⟹ (b) is a consequence of (ii), and (b) ⟹ (a) is trivial. From (i) and the inclusion S^{κ} \ κ(S) ⊆ M(S^{κ}) we infer that κ is surjective if I ≠ Ø, so (d) ⟹ (c). Since S does not contain any idempotent except 1 we also have (a) ⟹ (d).

The following corollary is only a handy reformulation of some of the assertions in 6.1, its proof is therefore omitted.

6.4. Corollary. Let S be a closed connected proper submonoid of Sl_{2} with dense interior and let (S^{κ}, κ) be a (topological) semigroup compactification of S. Then the following assertions are equivalent and imply that S^{κ} contains a zero element:

(i) The compactification map κ: S → S^{κ} is not injective.

(ii) There exists a closed (nonvoid) ideal I of S such that κ is constant on I.

(iii) κ: S → S^{κ} is surjective.

(iv) κ(S) meets the minimal ideal M(S^{κ}).

6.5. Aliens. Recall from [3] that an element s in a locally compact semigroup S is called an alien if it is the limit of a net ⟨x_{n}y_{n}⟩, where x_{n} → ∞ or y_{n} → ∞. The set of all aliens in S is denoted by Al(S). A locally compact topological semigroup is perfect if and only if it contains no aliens. If z is an alien then for any compactification map κ the image κ(z) is a large element in the sense of Section 5.9.

6.6. Corollary. Let S be a closed connected submonoid of Sl_{2} with dense interior. Then S contains aliens if and only if none of its compactification maps is injective. Equivalently: S is perfect if and only if there exists a compactification (S^{κ}, κ) such that κ(S) does not meet the minimal ideal M(S^{κ}).

Proof. This follows from 5.10 and 6.4.

6.7. Corollary. Let S be a closed subsemigroup of Sl_{2} with dense and connected interior. Then the following assertions are equivalent:

(i) S^{b} = M(S^{b}).

(ii) All elements of S are aliens in S.

Proof. The assertion is trivial in the case S = Sl_{2} , so we assume S ≠ Sl_{2} .

(i) ⟹ (ii) By [3], 5.2, 5.5, and 5.6 (cf. also 6.9 below), we know that the large ideal compactification maps the non-aliens of S onto non-minimal elements, so (i) implies (ii).

(ii) ⟹ (i) Note first that by definition, every point s ∈ S is the limit of products s_{n}t_{n} with lim s_{n} = lim t_{n} = ∞. Applying this fact to the factors s_{n} we see that every s ∈ S is also the limit of triple products x_{n}y_{n}z_{n} with lim x_{n} = lim y_{n} = lim z_{n} = ∞. Passing to suitable subnets we enforce that the limits u = lim b_{S}(x_{n}), v = lim b_{S}(y_{n}), and w = lim b_{S}(z_{n}) exist. We conclude that b_{S}(s) = uvw ∈ S^{b}vS^{b}, hence, by Theorem 5.8, S^{b} = M(S^{b}).

6.8. Example. (Cf. [3], 7.6) The set

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

is a closed subsemigroup of Sl_{2} with dense and connected interior, and all of its elements are aliens, thus, by the above corollary, its Bohr compactification is a paragroup. In fact, its Bohr compactification is a singleton set: We have seen in [3],7.6 that for every element s in a dense subset of the semigroup and all n ∈ ℕ there exist elements y_{n}, z_{n} in the semigroup such that

s = lim ( \begin{matrix}n & 0 \\ 1/n & 1/n \end{matrix} ) y_{n} = lim z_{n} ( \begin{matrix}1/n & 0 \\ n & n \end{matrix} ) .

This means that the Bohr compactification map maps s onto e.s = s.f, where e and f are fixed idempotents. It follows that the Bohr compactification of our semigroup is a left group as well as a right group, hence is a group. But since Sl_{2} is simple and noncompact this group must be singleton, by 6.2.

6.9. Large ideals. The following definitions and facts are taken from [3], section 5.

(i) Let S be a noncompact locally compact topological semigroup. An ideal I of S is said to be a large ideal if \overline{S \ I} is compact. Note that for closed large ideals I the Rees quotient S/I is always a compact topological semigroup.

(ii) If the intersection of all closed large ideals in S is nonempty then it is an ideal in S. If S is a subsemigroup of a topological group (more generally, if S is c-perfect ([3], 2.4)) then every element in this intersection is an alien.

(iii) Every non-alien element in a noncompact locally compact subsemigroup of a topological group G has a neighborhood which is the complement of a large ideal.

6.10. Large ideal compactifications. In the following assertions (i)– (iii) we let S be a noncompact locally compact subsemigroup of a topological group G (or, more generally, a noncompact c-perfect locally compact topological semigroup).

(i) Let A be a (possibly void) subset of S. Then the set I_{A} of all closed large ideals of S containing A is directed under ⊇ since for I, J ∈ I_{A} the intersection I ∩ J also lies in I_{A}. The intersection ⋂ I_{A} is the closed ideal generated by A ∪ Al(S).

(ii) The limit S^{ℓ_{A}} of the projective system {S/I}_{I∈I_{A}} exists and is a compact semigroup (cf. Theorem 2.22 of [6]), we write ℓ_{A} for the corresponding morphism S → S^{ℓ_{A}}. If A is empty then we call the ensuing compactification the (universal) large ideal compactification of S and denote it by (S^{ℓ}, ℓ) or S^{ℓ} for short.

(iii) The compactifications S^{ℓ_{A}} with A ≠ Ø are in 1-1-correspondence with the closed ideals of S which contain Al(S). In particular our discussion shows that for every closed ideal I of S containing Al(S) there exists a compactification (S^{κ}, κ) such that the restriction of κ to I is constant and the restriction of κ to S \ I is a topological embedding.

6.11. Notation. (i) We write NONINJ(S) for the set of equivalence classes (S^{κ}, κ) of compactifications of S with κ noninjective, augmented by the large ideal compactification (S^{ℓ}, ℓ). (Note that ℓ is injective if and only if Al(S) = Ø.) We provide NONINJ with the usual order of compactifications.

(ii) We write ALID(S) for the set of all closed ideals of S which contain Al(S), augmented by the empty set if Al(S) = Ø. The set ALID(S) is endowed with the usual order ⊆.

6.12. Theorem. Let S be a closed connected submonoid of Sl_{2} with dense interior. Then the following assertions hold:

(i) The map NONINJ(S) → ALID(S), assigning to (S^{κ}, κ) the set I = κ^{−1}(M(S^{κ})), is an order anti-isomorphism.

(ii) If Al(S) ≠ Ø then the Bohr compactification of S is equivalent with the

large ideal compactification. Note that the ordered sets NONINJ(S) and ALID(S) form complete lattices.

Proof. (i) follows from 6.4 and the remarks in 6.10.

(ii) The assertion trivially holds if S = Sl_{2} . So let us suppose that S ≠ Sl_{2} . We already know by 5.10 that S^{b} = b_{S}(S) ∪ M(S^{b}) and that b_{S}(Al(S)) ⊆ M(S^{b}). Since Al(S) is nonvoid we know by 6.1 and 6.6 that M(S^{b}) is a singleton. So the natural morphism S^{b} → S^{ℓ} separates the points of S^{b} and the assertion follows.

7. Idempotents and Directions

7.1. We start this section with the observation that the map of 5.1(v), assigning to every X ∈ rlog(S ∩ exp(Kill^{+})) the idempotent e_{X}, is continuous and can be extended continuously to the set D = \overline{rlog(S ∩ exp(Kill^{+}))} of regular directions in S. A further extension, dealing also with ‘non-regular directions,’ will be given in 7.5 below.

7.2. Proposition. Assume that S is a closed connected proper submonoid of Sl_{2} with dense interior. Then there exists a unique continuous map ε: D → E(M(S^{κ})) with the following property (a):

(a) If ⟨x_{i}⟩ is a net in S ∩ exp(Kill^{+}) with

lim x_{i} = ∞, lim rlog(x_{i}) = X, lim κ(x_{i}) = m ∈ S^{κ},
then m ∈ H(ε(X)).

Proof. We first pick a point X ∈ D and claim that there exists a unique idempotent ε(X) ∈ M(S^{κ}) such that (a) holds.

The definition of Umb immediately implies that there exists a net ⟨x_{i}⟩ in S ∩ exp(Kill^{+}) such that lim x_{i} = ∞, lim rlog(x_{i}) = X, we may also assume that lim κ(x_{i}) = m ∈ S^{κ} exists. By 5.10, m ∈ M(S^{κ}), so there exists a minimal idempotent ε(X) with m ∈ H(ε(X)). To show that the definition of ε(X) does not depend on the choice of ⟨x_{i}⟩, suppose that ℓ = lim κ(y_{j}) for another net ⟨y_{j}⟩ in S ∩exp(Kill^{+}) with lim y_{j} = ∞, lim rlog(y_{j}) = X. Then, by 5.10, ℓ ∈ M(S^{κ}), so we have to prove that ℓ ∈ mS^{κ}m. Pick (a, b) ∈ Int S × Int S such that a^{∘}X^{∘}b ∈ Int D, the set of all such pairs is dense in S × S (cf. 3.9(i),(ii)). By the asymptotic formulas 3.8 and by Lemma 5.4, we have {a.ℓ.b, a.m.b} ⊆ H(e_{a^{∘} X^{∘} b}) and therefore a.ℓ.b ∈ (a.m.b)S^{κ}(a.m.b). Taking limits a → 1, b → 1 proves our claim.

We now show that ε is continuous. Consider a net ⟨X_{i}⟩ of Umb(Int S)∩ Hyp converging to X ∈ D. Then there exists a net ⟨t_{i}⟩ of positive reals with t_{i} → ∞ and such that exp(t_{i}X_{i}) ∈ Int S for every index i. Clearly, lim exp(t_{i}X_{i}) = ∞ and we have

ε(X_{i}) = e_{X_{i}} ∈ exp(t_{i}X_{i}).H(e_{X_{i}}). exp(t_{i}X_{i}).

Then, by (a), every limit of a convergent subnet of the net ⟨e_{X_{i}}⟩ must be equal to ε(X), hence every convergent subnet of ⟨e_{X_{i}}⟩ has the limit ε(X). This shows that lim ε(X_{i}) = ε(X). The asserted continuity now follows from the fact that Umb(Int S) ∩ Hyp is dense in D.

7.3. Basic properties of the map ε. We retain the assumptions of 7.2.

(i) If x = exp(tX) ∈ S with X ∈ Hyp, t > 0, then X ∈ D and e_{X} = ε(X).

(ii) If two elements X, X′ of D lie on the same horizontal [vertical] line then the corresponding idempotents ε(X), ε(X′) lie in the same minimal right [ left ] ideal of S^{κ}.

(iii) If X, Y ∈ D and the diamond products X^{♢}Y and Y ^{♢}X exist then ε(X)ε(Y) ∈ H(ε(X^{♢}Y)), or, equivalently, ε(X^{♢}Y)ε(Y) = ε(X)ε(X^{♢}Y) = ε(X^{♢}Y).

(iv) If X ∈ D and s ∈ S then s.ε(X) ∈ H(ε(s^{∘}X))) and ε(X).s ∈ H(ε(X^{∘}s))).

(v) If X ∈ D then vert(X) ∩ D = \overline{S^{∘}X}, and E(S^{κ}ε(X)) = \overline{ε(vert(X) ∩ D)}. Similarly, hor(X) ∩ D = \overline{X^{∘}S}, and E(ε(X)S^{κ}) = \overline{ε(hor(X) ∩ D)}.

(vi) If X is an interior point of D in Hyp then the sets hor(X) ∩ D and vert(X) ∩ D are compact line segments and the map

\begin{matrix}(vert(X) ∩ D) × (hor(X) ∩ D) → E(M(S^{κ})), \\ (A, B) ↦ (ε(A)ε(B))^{−1}ε(A)ε(B), \end{matrix}

is continuous and surjective. (vii) If ε(X) = ε(Y) then for every Z ∈ D we also have ε(X^{♢}Z) = ε(Y ^{♢}Z)

and ε(Z^{♢}X) = ε(Z^{♢}Y), provided that the diamond products exist. (viii) If ε(X) = ε(Y) then for every s ∈ S we also have ε(s^{∘}X) = ε(s^{∘}Y)

and ε(X^{∘}s) = ε(Y ^{∘}s).

Proof. Assertion (i) is left to the reader, (ii) follows from 5.3(iv) since ε is continuous, and (iii) is a direct consequence of (ii) and the definition of the ♢ -product.

(iv) Let ⟨x_{i}⟩ be a net in S∩exp(Kill^{+}) such that x_{i} → ∞, rlog(x_{i}) → X, and lim κ(x_{i}) = ε(X). Then rlog(sx_{i}) → s^{∘}X and rlog(x_{i}s) → X^{∘}s, by 3.8, so the assertion follows now from 7.2.

(v) The assertions 3.7(i) and 3.9(ii) yield that vert(X) ∩ D = \overline{S^{∘}X} and hor(X) ∩ D = \overline{X^{∘}S}. Combined with (iv), these equalities finally imply that E(S^{κ}ε(X)) = \overline{ε(vert(X) ∩ D)} and E(ε(X)S^{κ}) = \overline{ε(hor(X) ∩ D)}.

(vi) The compactness of hor(X) ∩ D and vert(X) ∩ D follows from the classification of rectangular domains in section 7 and example 5.3 of [2], it is also not difficult to devise a direct proof. Now (v) implies that E(S^{κ}ε(X)) = ε(vert(X) ∩ D) and E(ε(X)S^{κ}) = ε(hor(X) ∩ D), and therefore the continuity as well as the surjectivity of the map in (v) follows from the general theory of compact topological semigroups and the continuity of ε.

Assertion (vii) is an immediate consequence of (iii), (viii) follows from 7.4. A useful formula. We retain the assumptions of 7.2 and suppose that the matrix H is an interior point of D in Hyp. We write k for the unique continuous homomorphism ℝ → H(ε(H)) such that k(t) = ε(H). exp(tH) whenever exp(tH) ∈ S (cf. 5.3(iii)). Then for every matrix s = \binom{a b}{c d} ∈ S we have the formula

ε(H)κ(s)ε(H) = k(log(a)).

This formula will be applied in 7.9.

Proof. Let ⟨t_{i}⟩ be a net in ℝ^{+} with t_{i} → ∞ and lim κ(exp(t_{i}H)) = ε(H). Then

(∗) exp(t_{i}H)s exp(t_{i}H) = ( \begin{matrix}1 & 0 \\ \frac{c}{a}e^{−2t_{i}} & 1 \end{matrix} ) ( \begin{matrix}ae^{2t_{i}} & 0 \\ 0 & \frac{e^{−2t_{i}}}{a} \end{matrix} ) ( \begin{matrix}1 & \frac{b}{a}e^{−2t_{i}} \\ 0 & 1 \end{matrix} ) .

For any inner point u of S and all sufficiently large indexes i the points

u ( \begin{matrix}1 & 0 \\ \frac{c}{a}e^{−2t_{i}} & 1 \end{matrix} ) , ( \begin{matrix}ae^{2t_{i}} & 0 \\ 0 & \frac{e^{−2t_{i}}}{a} \end{matrix} ) = exp((2t_{i} + log(a))H), ( \begin{matrix}1 & \frac{b}{a}e^{−2t_{i}} \\ 0 & 1 \end{matrix} ) u

are contained in S. Note that ε(H)κ(exp((2t_{i} + log(a))H)) = k(2t_{i} + log(a)). Now, applying κ to (∗) and passing to limits, we see that u.ε(H)κ(s)ε(H).u = u.ε(H)k(log(a))ε(H).u. Since u can be chosen arbitrarily near the identity this implies the assertion.

The next proposition shows that ε is compatible with the map c of 3.2(iv), in that it can be extended to a continuous map on the set of asymptotic directions Asy(S) = \overline{c(D)}.

7.5. Proposition. Under the assumptions of 7.2 we always have a continuous and surjective map ε̄: Asy(S) → E(M(S^{κ})) which extends ε, that is, ε̄(c(X)) = ε(X), for all X ∈ D. This map has the following properties:

(i) If ⟨x_{i}⟩ is a net in S ∩ exp(Kill^{+}) with

lim x_{i} = ∞, lim c(rlog(x_{i})) = (h, v), lim κ(x_{i}) = m ∈ S^{κ},

then m ∈ H(ε̄(h, v)). (ii) If (h, v), (h′, v′) ∈ \overline{c(D)} then ε̄(h, v)ε̄(h′, v′) ∈ H(ε̄(h, v′)). (iii) ε̄ is injective if and only if ε is injective.

Proof. We pick an interior point Z ∈ D and claim that \overline{c(D)} = hor(D^{♢}Z) × vert(D^{♢}Z). Indeed, for X ∈ D we have hor(X) = hor(X^{♢}Z) and vert(X) = vert(Z^{♢}X), thus \overline{c(D)} ⊆ hor(D^{♢}Z) × vert(D^{♢}Z). On the other hand, if (h, v) ∈ hor(D^{♢}Z) × vert(D^{♢}Z) with h ∩ v ≠ Ø then (h, v) ∈ c(D). Since h ∩ v = Ø implies that neither h nor v can meet the interior of D, we conclude that c(D) misses at most two points of hor(D^{♢}Z) × vert(D^{♢}Z). (In fact, the missing points must be corner points.)

We define ε̄(h, v) = e, where e is the unique idempotent in the H-class of ε(U)ε(V) and h ∩ vert(Z) = {U}, v ∩ hor(Z) = {V }. Obviously, ε̄ extends ε. By Corollary 7.3(vi), ε̄ is continuous and surjective. By continuity, assertion (ii) follows from 7.3(iii).

(i) Let ⟨x_{i}⟩ be a net with the properties of (i). Then we may and do assume that the limits A = lim rlog(x_{i})^{♢}Z and B = lim Z^{♢} rlog(x_{i}) exist, note that then h ∩ vert(Z) = {A} and hor(Z) ∩ v = {B}. Now, by 7.3(iii), ε(rlog(x_{i})^{♢}Z) = ε(rlog(x_{i}))ε(rlog(x_{i})^{♢}Z) ∈ κ(x_{i})S^{κ}, so, by continuity, ε(A) ∈ mS^{κ}. In the same way we see that ε(B) ∈ S^{κ}m, thus ε(A)ε(B) ∈ mS^{κ} ∩S^{κ}m = H(m), which implies m ∈ H(ε(A)ε(B)) = H(ε̄(h, v)).

(iii) It is obvious that ε must be injective if ε̄ is injective Suppose now that ε is injective but ε̄ is not. Then we have ε̄(h, v) = ε̄(h′, v′), for some (h, v) ∈ \overline{c(D)} \ c(D), (h′, v′) ∈ \overline{c(D)}. Let X be an inner point of D in Hyp. If h ≠ h′ then (h, vert(X)) and (h′, vert(X)) are distinct and lie in c(D), so ε̄(h, vert(X)) ≠ ε̄(h′, vert(X)), and (ii) implies a contradiction. Similarly, if v ≠ v′ then ε̄(hor(X), v) ≠ ε̄(hor(X), v′) leads to a contradiction.

For the next result recall that a semigroup is called left [ right ] simple if it contains no proper left [ right ] ideals.

7.6. Theorem. Let (S^{κ}, κ) be a topological semigroup compactification of a closed connected proper submonoid S of Sl_{2} with dense interior, and suppose that the map ε: D → E(M(S^{κ})) of 7.2 is not injective. Then the minimal ideal of S^{κ} is either left or right simple and all of its maximal subgroups are singleton. More specifically, the following assertions are equivalent:

(i) ε(A) = ε(B) for two distinct elements A, B ∈ D which lie on the same horizontal [ vertical ] line;

(ii) ε is constant along any horizontal [ vertical ] line, i.e., ε(hor(X) ∩ D) [ ε(vert(X) ∩ D) ] is singleton for every X ∈ D;

(iii) the minimal ideal M(S^{κ}) is a minimal left [ right ] ideal.

Proof. If ε(A) = ε(B) then ε(A) = ε(A^{♢}B) = ε(B) = ε(B^{♢}A), provided that the diamond products exist (by 7.3(vii)), thus if ε is not injective on D then it is not injective either on a horizontal line or on a vertical line. We only treat the case where A, B lie on the same horizontal line.

If (iii) holds then for every idempotent e ∈ M(S^{κ}) the map S → H(e), s ↦ e.s.e is a continuous homomorphism mapping S onto a dense subsemigroup of the compact topological group H(e), so H(e) = {e}, by 6.2.

(i) ⟹ (ii) We assume that X lies in the interior of D and that ε(X) = ε(Y) for some Y ∈ D∩hor(X)\{X}. By 7.3(vii),(viii) our assumption (i) implies that such pairs X, Y exist.

Since X is in the interior of D there exists a positive real T such that exp(tX) ∈ S whenever t ≥ T. We know from 3.9(ii) that lim_{t→∞} Y ^{∘} exp(tX) = Y ^{♢}X = X. By 3.9(i) and 7.3(viii), ε(X) = ε(X^{∘} exp(tX)) = ε(Y ^{∘} exp(tX)), for all t ≥ T, therefore ε must be constant on the set X ∪ Y ^{∘} exp([T, ∞[·X)), which is a non-degenerate line segment joining X and Y ^{∘} exp(T X) (see 3.9(ii)). Thus the ε-class ε^{−1}(ε(X))∩hor(X) of X has interior points in hor(X). Now we observe that ε^{−1}(ε(X^{∘}s)) ⊇ ε^{−1}(ε(X))^{∘}s, so ε^{−1}(ε(X′)) ∩ hor(X) has nonvoid interior in hor(X), for every X′ ∈ X^{∘}S (recall that ^{∘} is the restriction of a group action). Since hor(X) is separable this means that ε assumes at most countably many different values on X^{∘}S, since X^{∘}S is pathwise connected we conclude that ε is constant on X^{∘}S. But X^{∘}S is dense in hor(X) ∩ D (by 7.3(v)), so ε must be constant on hor(X) ∩ D. For arbitrary Z ∈ D we have hor(Z) ∩ D = Z^{♢}(hor(X) ∩ D), so ε is constant also on hor(Z) ∩ D, in view of 7.3(vii).

(ii) ⟹ (iii) Since for any X in the interior of D the map ε sends (hor(X) ∩ D) onto the set of all idempotents of the minimal right ideal ε(X)S^{κ} (by 7.3(v),(vi)), the assertion follows from the general structure theorem about the minimal ideal in a compact topological semigroup.

The implication (iii) ⟹ (i) follows from 7.3(ii).

7.7. Corollary. If there exist elements X, Y ∈ D, not lying in the same Borel algebra, such that ε(X) = ε(Y) then S^{κ} has a zero element.

Proof. This follows from 7.6 and the fact that the four points X, X^{♢}Y , Y , Y ^{♢}X are distinct if X and Y do not lie in the same Borel subalgebra.

7.8. Notation. Suppose now that D is compact (or, equivalently, that \overline{Umb}(S) does not contain any nonzero nilpotent elements, cf 4.2). Let Σ = exp(ℝ_{0}^{+}D) and consider the maps i_{S}: S → \overline{i(S)} and j_{S}: S → \overline{j(S)} defined in 4.16. We abbreviate \overline{i(S)} to S^{D}, and \overline{j(S)} to S^{M}. Recall that by 4.17 the compactification S^{D} is equivalent with the ⊔-product S ⊔_{φ′} D, and that, by 4.19, the compactification S^{M} is equivalent with S ⊔_{Φ′} M(D, K).

For the next theorem we define the following congruences L_{M}, R_{M} on S^{D}, which are derived from Green’s relations:

aL_{M}b if \{ \begin{matrix}a = b & or \\ a, b ∈ D & and ab = a; \end{matrix}

aR_{M}b if \{ \begin{matrix}a = b & or \\ a, b ∈ D & and ab = b. \end{matrix}

Note that the join L_{M} ∨R_{M} is the congruence which collapses the minimal ideal D of S^{D} to a zero element.

7.9. Theorem. Let S be a closed connected proper submonoid of Sl_{2} with dense interior such that the set D of regular directions is compact, and let (S^{κ}, κ) be a topological semigroup compactification of S with injective compactification morphism κ. We suppose that the interior of D in Hyp contains the point H (this can be always enforced by applying a suitable inner automorphism). We write k for the homomorphism, constructed in 5.3(iii), ℝ → K = H(e_{H}) with k(t) = e_{H}. exp(tH) whenever exp(tH) ∈ S. Then the following assertions hold:

(i) If the map ε: D → E(M(S^{κ})), defined as in 7.2, is not injective then k is constant and S^{κ} is isomorphic to S^{D}/ρ, where ρ is one of the congruences L_{M}, R_{M} or L_{M} ∨ R_{M}.

(ii) If ε is injective then S^{κ} is isomorphic to S^{M}, where M = M(D, K). An isomorphism is given by the map μ: S^{M} = S ∪ M(D, K) → S^{κ}, where μ(s) = κ(s) if s ∈ S and μ(X^{♢}H, g, H^{♢}X) = ε(X^{♢}H)gε(H^{♢}X) if X ∈ D, g ∈ K. The compactification (S^{κ}, κ) is equivalent with the Bohr compactification if and only if (K, k) is equivalent with the Bohr compactification of ℝ.

Proof. Assertion (i) follows from Theorem 7.6 and Corollary 7.7.

(ii) We first show that the map μ is a continuous homomorphism. Since the restriction of μ to S is obviously continuous, it suffices to show that κ(s_{n}) → ε(X^{♢}H)gε(H^{♢}X) whenever

⟨s_{n} = ( \begin{matrix}a_{n} & b_{n} \\ c_{n} & d_{n} \end{matrix} )⟩

is a net in S such that lim κ(s_{n}) exists and

(†)  s_{n} → (X^{♢}H, g, H^{♢}X) in S^{M}.

The definition of the topology of S^{M} implies that (†) is equivalent to the following condition:

s_{n} → ∞, rlog(s_{n}) → X, k(log(a_{n})) → g. Applying formula 7.4 and taking limits we see that

lim ε(H)κ(s_{n})ε(H) = lim ε(H)k(log(a_{n}))ε(H) = g.

Let ⟨κ(s_{i})⟩ be a convergent subnet of ⟨κ(s_{n})⟩. Since rlog(s_{i}) → X we have (recall 7.2 and 7.3(iii))

lim κ(s_{i}) = lim ε(X^{♢}H)κ(s_{i})ε(H^{♢}X) = lim ε(X^{♢}H)ε(H)κ(s_{i})ε(H)ε(H^{♢}X) = ε(X^{♢}H)gε(H^{♢}X) .

So every convergent subnet of ⟨κ(s_{n})⟩ converges to ε(X^{♢}H)gε(H^{♢}X), thus ⟨κ(s_{n})⟩ converges itself to this element. It follows that μ is continuous, hence it is also a homomorhism (being a homomorphism on the dense subset S).

By definition, the map μ is injective on S as well as on M = M(D, K), so by 6.1 it is injective on S^{M}. The surjectivity follows by definition (or since κ(S) is dense in S^{κ}).

The rest of the assertion follows from the universality of the Bohr compactification of ℝ.

7.10. Corollary. We retain the assumptions and the notation of the above Theorem 7.9. Let X = H + βP ∈ hor(H) ∩ D and Y = H + γQ ∈ vert(H) ∩ D. Then ε(X)ε(Y) = k(log(1 + βγ/4)).

Proof. This formula follows from Theorem 7.9 and the definition of S^{M} (7.8, 4.10).

7.11. Proposition. Let S be a closed connected proper submonoid of Sl_{2} with dense interior and assume that D is not compact. Assume also that the Lie wedge of S contains a non-nilpotent element. (This assumption is automatically satisfied if S is a Lie semigroup.) Then the minimal ideal of any compactification S^{κ} of S consists of idempotents.

Proof. If ε fails to be injective then the assertion follows from 7.6. Suppose now that ε is injective.

Applying a suitable inner automorphism of 𝔰𝔩(2, ℝ) we enforce that H lies in the interior of D in Hyp. Since D is noncompact there exist X = H + γQ ∈ D ∩ vert(H) and Y = H + βP ∈ D ∩ hor(H) such that X^{♢}Y does not exist, or, equivalently, such that βγ = −4 (cf. [2] 4.15). We know that the ♢ product A^{♢}B always exists if one of A, B lies in the interior of D (in Hyp). Thus X and Y must lie on the boundary of D in Hyp, so X is an endpoint of vert(H) ∩ D and Y is an endpoint of hor(H) ∩ D. Let X′ = H + γ′Q be the endpoint ≠ X of vert(H) ∩ D, and Y′ = H + β′P the endpoint ≠ Y of hor(H) ∩ D.

By assumption we can find an element Z of D which also lies in the Lie wedge of S. Since X^{♢}Y does not exist, Z cannot live both in hor(X) and in vert(Y). Similarly, if also X′^{♢}Y′ does not exist then Z ∉ hor(X′) ∩ vert(Y′). We assume that Z ∉ hor(X) and Z ∉ vert(Y′), the other cases can be treated analogously and are therefore left to the reader. For 0 < η ≤ 1 we next define

X_{η} = H + (1 − η)γQ, Y′_{η} = \{ \begin{matrix}Y′ & if X′^{♢}Y′ exists \\ H + (1 − η)β′P & otherwise \end{matrix}

and note that the four elements X_{η}, X′, Y, Y′_{η} generate a compact ^{♢} -subse-migroup D_{η} of D. Thus exp(ℝ_{0}^{+}D_{η}) is an exponential subsemigroup of Sl_{2} . Moreover, if η is sufficiently small, say η ≤ η_{0} , then Z is contained in the interior of D_{η} in D, so S_{η} = S ∩exp(ℝ_{0}^{+}D_{η}) is a closed submonoid of Sl_{2} whose interior clusters at 1. Since S_{η} ⊆ \overline{exp(Kill^{+})} we conclude that S_{η} is connected ([2] 3.4(iv)). Clearly, our compactification (S^{κ}, κ) restricts to a topological semigroup compactification of S_{η}. Moreover, H(e_{H}) ⊆ \overline{κ(S_{η})}.

As in 7.9 we now put K = H(e_{H}) and write k for the induced compactification morphism ℝ → K. Pick any x_{0} ∈ K. We shall show that x_{0} = ε(Y)ε(X), since x_{0} was arbitrary this means that K is a singleton and the assertion follows.

The set k(] − ∞, log(η_{0})]) is dense in K, thus we can find a net ⟨η_{n}⟩ in ℝ with 0 < η_{n} ≤ η_{0} and lim η_{n} → 0, k(log(η_{n})) → x_{0} . Now by 7.10 k(log(η_{n})) = ε(H +βP)ε(H +(1−η_{n})γQ) → ε(Y)ε(X) which finishes our proof.

7.12. Example. (cf. [17] 4.1(ii), [2] 6.5, 10.12) Let S be the semigroup of all 2 × 2-matrices \binom{a b}{c d} with nonnegative entries in Sl_{2} and a ≥ 1. Then the Bohr compactification of S is equivalent with the one-point compactification S_{∞} = S ∪ {∞}.

Proof. We use our current notation for the special case κ = b_{S}. Recall that S is a perfect Lie semigroup with Lie wedge W = ℝ_{0}^{+}H + ℝ_{0}^{+}P + ℝ_{0}^{+}Q ([11], p. 419, [17] 4.1(ii), [3] 6.13) and that \overline{Umb}(S) = ℝH + ℝ_{0}^{+}P + ℝ_{0}^{+}Q ([2] 10.12). Since D = \overline{Umb}(S) ∩ Hyp is noncompact we conclude from 7.11 that the maximal subgroups in S^{κ} are singleton. Also, the set H + ℝ_{0}^{+}P is a horizontal line in D, the set H + ℝ_{0}^{+}Q is a vertical line in D. Now the relation exp(t(H + λP)) exp(sP) = exp(e^{2t}sP) exp(t(H + λP)), for every s, t ∈ ℝ^{+} , implies that e_{X} = e_{X}e_{P} = e_{P} e_{X} = e_{P} for every X ∈ H + ℝ_{0}^{+}P. Similarly, e_{Y} = e_{Q} for all Y ∈ H + ℝ_{0}^{+}Q, and the assertion follows from 7.7.

8. Summary of the main results

8.1. Theorem. Let (S^{κ}, κ) be a topological semigroup compactification of a closed connected proper submonoid S of Sl_{2} with dense interior. Then the following assertions hold:

(i) The S -large elements of S^{κ} lie in the minimal ideal M(S^{κ}). Thus, in particular, S^{κ} = κ(S) ∪ M(S^{κ}).

(ii) Let D = \overline{rlog(S ∩ exp(Kill^{+}))} and write Asy(S) for the set \overline{c(D)} of all asymptotic directions in S. Then there exists a continuous surjection ε̄: Asy(S) → E(M(S^{κ})) which is a homomorphism mod H, that is, ε̄(h, v)ε̄(h′, v′) ∈ H(ε̄(h, v′)).

(iii) Exactly one of the following cases takes place:

(a) ε̄ is injective, hence a homeomorphism. In this case κ is an embedding.

(b) ε̄(h, v) = ε̄(h′, v′) if and only if v = v′. Then M(S^{κ}) consists of right zeros of S^{κ}.

(c) ε̄(h, v) = ε̄(h′, v′) if and only if h = h′. Then M(S^{κ}) consists of left zeros of S^{κ}.

(d) M(S^{κ}) is singleton. In this case S^{κ} has a zero element.

(iv) Every maximal subgroup of S^{κ} is isomorphic with a compactification of ℝ.

Proof. Assertion (i) is part of Theorem 5.10 and (ii) is 7.5(i),(ii). Assertion (iii) follows from 7.5(iii), 6.4, and 7.6. The last assertion is a consequence of 5.3(ii),(iii).

8.2. Explicit Constructions. As before we let (S^{κ}, κ) be a topological semigroup compactification of a closed connected proper submonoid S of Sl_{2} with dense interior.

(i) If κ is not injective then there exists a closed ideal I of S such that (S^{κ}, κ) is equivalent with the projective limit lim_{←}(S/J, ℓ_{J}), where J runs over all closed large ideals of S which contain I, and for every J the map ℓ_{J} is the quotient morphism S → S/J.

(ii) If D is compact and ε̄ is injective then there exists a compactification (K, k) of ℝ and a conjugate S_{1} of S in Sl_{2} such that (S^{κ}, κ) is equivalent with a ⊔-product (S_{1}^{M} = S_{1} ⊔ M(D, K), j_{S_{1}}), as described in 4.19.

(iii) If D is compact, κ is injective, and ε̄ is not injective then (S^{κ}, κ) is equivalent with a quotient (S^{D}/ρ, i_{ρ}) of (S^{D}, i_{S}), where S^{D} = S ⊔ D is defined as in 4.17, and ρ is one of the congruences L_{M}, R_{M} or L_{M} ∨ R_{M} of 7.8.

Proof. (i) follows from 6.4 and 6.12, and (ii),(iii) are consequences of 7.9.

8.3. Remark. (i) The quotients S^{D}/ρ can be defined also as ⊔-products. (Exercise)

(ii) Note that D is compact if and only if S is contained in a perfect exponential subsemigroup of Sl_{2} . At present we do not have explicit constructions for the case where D is noncompact but S is perfect.

8.4. The lattice of all compactifications of S. Let S be a closed connected proper submonoid of Sl_{2} with dense interior.

(i) If D is compact then we have complete information about the lattice of all compactifications of S. As sketched in the diagram at the left this lattice consists of three parts:

(1) On top we have the sublattice of those compactifications where ε is injective. These compactifications are characterized by the associated compactification (K, k) of the reals, thus this part of the lattice is isomorphic with the lattice of all topological group compactifications of ℝ, which, in turn, is isomorphic with the lattice of all subgroups of the discretization ℝ_{d} of ℝ, ordered by A ≤ B if A ⊆ B. The minimal element in the lattice of injective compactifications is S^{D}.

(2) Next comes the diamond lattice made up of S^{D} and its quotients S^{D}/L_{M}, S^{D}/R_{M} and S^{D}/(L_{M} ∨ R_{M}) = S_{∞}.

(3) On the bottom we have the lattice formed by S_{∞} and all topological semigroup compactifications with noninjective compactification map. This lattice is isomorphic with the lattice formed by the closed ideals of S and the empty set, ordered by A ≤ B if A ⊇ B.

S^{b} = S^{M} [figure: ···; |; ···]_{· · · · · · · · · · · · · · · · · · · · · · · ·} [figure: ···; |; ···] S^{D} [figure: S^{D}/L_{M}; S^{D}/R_{M}] S_{∞} = S ∪ {∞} [figure: ···; |; ···]_{· · · · · · · · · · · · · · · · · · · · · · · ·} [figure: ···; |; ···] {0}

(ii) If S is perfect but D is noncompact then the compactifications described in (3) above form a full sublattice of the lattice of all topological semigroup compactifications. At the moment we do not know whether this sublattice is a proper sublattice. If the Lie wedge of S contains at least one regular matrix then by 7.11 all maximal subgroups of S^{b} are trivial, so part (1) of (i) has no counterpart in this case. (iii) If S is not perfect then we know from 6.6 and 6.12 that the compactification lattice of S is anti-isomorphic with the ⊆-lattice ALIDS of all closed ideals containing the aliens of S. (iv) If S is an exponential semigroup then D is compact if and only if S is perfect if and only if the Lie wedge of S contains no nonzero nilpotent elements. Thus for exponential semigroups S with inner points in Sl_{2} there are exactly two cases: (a) S is perfect— then D is compact and the compactification lattice has the form described in (i) above; (b) S is nonperfect— then the compactification lattice is as described in (iii). (Al(S) has been explicitly calculated in [3].)

References

[1] Bourbaki, N., “General Topology I,” Hermann, Paris 1966.

[2] Breckner, B. E., and W. A. F. Ruppert, On Asymptotic Behavior and Rectangular Band Structures in Sl(2, ℝ), J. Lie Theory 11 (2001), 559– 604.

[3] —, Perfect Semigroups and Aliens, Semigroup Forum 66 (2003), 236– 272.

[4] —, A note about congruences on subsemigroups of topological groups (to appear in Semigroup Forum).

[5] Brown, D. R., and M. Friedberg, Linear representations of certain compact semigroups, Trans. Amer. Math. Soc. 160 (1971), 453–465.

[6] Carruth, J. H., J. A. Hildebrant, and R. J. Koch, “The Theory of Topological Semigroups,” Marcel Dekker, Inc., New York and Basel, 1983.

[7] Engelking, R., “General Topology,” Warsaw 1977.

[8] Friedberg, M., Almost periodic functions, compactifications, and faces of finite dimensional cones, Math Z. 176 (1981), 53–61.

[9] —, Compactifications of finite-dimensional cones, Semigroup Forum 15 (1978), 199–228 .

[10] Hewitt, E., and K. Ross, “Abstract Harmonic Analysis I,” Springer Grundlehren 115, Berlin etc., 1963.

[11] Hilgert, J., K. H. Hofmann, and J. D. Lawson, “Lie Groups, Convex Cones and Semigroups,” Oxford University Press, 1989.

[12] Hofmann, K. H., and W. A. F. Ruppert, “Lie Groups and Subsemigroups with Surjective Exponential Function,” Memoirs of the AMS 130 (1997), No. 618.

[13] Jonsdottir, K. H., Ideal compactifications in uniquely divisible semigroups, Semigroup Forum 14 (1977), 355–374.

[14] —, Kernel compactifications of uniquely divisible semigroups, Semigroup

Breckner and Ruppert 109

Forum 15 (1978), 269–279.

[15] Ruppert, W. A. F., “Compact Semitopological Semigroups: An Intrinsic Theory,” Springer-Verlag, 1984.

[16] —, A Geometric Approach to the Bohr Compactification of Cones, Math. Z. 199 (1988), 209–232.

[17] —, Bohr compactifications of subsemigroups of nonabelian Lie groups, Semigroup Forum 37 (1988), 325–342.

B. E. 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 October 1, 2002 and in final form April 24, 2003