Journal of Lie Theory Volume 14 (2004) 317–338 \begin{matrix}c \\ ◯ \end{matrix} 2004 Heldermann Verlag

Tits Geometry, Arithmetic Groups, and the Proof of a Conjecture of Siegel

Enrico Leuzinger

Communicated by G. A. Margulis

Abstract. Let X = G/K be a Riemannian symmetric space of noncompact type and of rank ≥ 2. An irreducible, non-uniform lattice Γ ⊂ G in the isometry group of X is arithmetic and gives rise to a locally symmetric space V = Γ\X. Let π : X → V be the canonical projection. Reduction theory for arithmetic groups provides a dissection V = ∐_{i=1}^{k} π(X_{i}) with π(X_{0}) compact and such that the restiction of π to X_{i} is injective for each i. In this paper we complete reduction theory by focusing on metric properties of the sets X_{i}. We detect subsets C_{i} of X_{i} (ℚ–Weyl chambers) such that π_{|C_{i}} is an isometry and such that C_{i} is a net in X_{i}. This result is then used to prove a conjecture of C.L. Siegel. We also show that V is quasi-isometric to the Euclidean cone over a finite simplicial complex and study the Tits geometry of V .

1. Introduction and survey of results

In this paper we study geometric properties of certain locally symmetric spaces V = Γ\X. Here X is a Riemannian symmetric space of noncompact type (without Euclidean factor) and Γ is a non-uniform (torsion-free) lattice in the group of isometries of X, i.e., the quotient space V is not compact but has finite volume. We make the additional assumption that the rank of X is at least two and we also assume that the non-uniform lattice Γ is irreducible. In that case a fundamental theorem of G.A. Margulis asserts that Γ is arithmetic. Roughly this means that there is a semisimple algebraic group G ⊂ GL(n, ℂ) defined over ℚ such that Γ is commensurable with G(ℤ) = G ∩ GL(n, ℤ) (see [30] Ch. 6 and Section 2 below). The symmetric space X can be written as the homogeneous space G(ℝ)^{0}/K for a (chosen) maximal compact subgroup K. The G(ℝ)^{0} -invariant Riemannian metric on X induces one on V so that the canonical projection π : X ⟶ Γ\X is a Riemannian covering.

1.1. Metric properties of locally symmetric spaces. Fundamental domains provide a basic and classical tool to investigate quotients of discrete groups acting isometrically on spaces of constant curvature. For lattices in

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag isometry groups of higher rank symmetric spaces exact fundamental domains have a rather complicated geometry. In order to study locally symmetric spaces V = Γ\X one therefore uses only coarse fundamental domains, whose shapes reflect certain (algebraic) decompositions of the isometry group of X.

Reduction theory provides such coarse fundamental domains Ω ⊂ X for arithmetic groups. Its basic building blocks are so-called Siegel sets, which in turn are unions of certain Euclidean cones (ℚ–Weyl chambers). A more refined version, precise reduction theory, yields a combinatorial dissection of V into disjoint subset: V = ∐_{i=0}^{l} V_{i} where V_{0} is compact and the other V_{i} are noncompact and can be explicitely described as one-to-one images of Siegel sets, V_{i} = π(S_{i}), where π : X ⟶ V = Γ\X denotes the canonical projection map. A variant of such a dissection based on an exhaustion of V = Γ\X by compact submanifolds with corners has been created in [23] and in [24] and will be our main tool in investigating V . We shall discuss reduction theory and exhaustions in more detail in Sections 2 and 3.

Our main result (Theorem 4.1) completes precise reduction theory by revealing metric properties of subsets of the above V_{j}:

Theorem A. There are ℚ–Weyl chambers C_{i} ⊂ S_{i}, 1 ≤ i ≤ l, in X such that, for each i, the restriction of the π : X ⟶ V to C_{i} is an isometry (with respect to the distance functions induced by the Riemannian metrics on X and V , respectively). Moreover ⋃_{i=1}^{l} π(C_{i}) is a net in V .

As a consequence of Theorem A we confirm a conjecture of C.L.Siegel.

Theorem B. The projection map π : X ⟶ Γ\X restricted to a Siegel set is a (1, D)−quasi-isometry.

Siegel’s conjecture has been proved independently by L. Ji and in the special case of SL(n, ℤ) ⊂ SL(n, ℝ) also by J. Ding (see [18] and [11]). It has been applied recently by D. Kleinbock and G.A. Margulis to investigate logarithm laws for certain flows on locally symmetric spaces (see [21]).

1.2. Quasi-isometry invariants for locally symmetric spaces. Using the assertion of Theorem A we construct a quasi-isometric model of the locally symmetric space V = Γ\X. It is obtained as the asymptotic cone which is defined as the Gromov-Hausdorff limit of “rescaled” pointed metric spaces:

Cone(V) := H − lim_{n→∞}(V, v_{0}, \frac{1}{n}d_{V}),

where v_{0} is an arbitrarily chosen point of V and d_{V} is the distance function on V induced by the Riemannian metric (see e.g. [13] Ch. 3 or [10] Ch. I). We remark that in contrast to the case considered here the definition of an asymptotic cone in general involves the use of ultrafilters, and the limit space may even depend on the chosen ultrafilter. Various aspects of asymptotic cones of general spaces are discussed in M. Gromov’s essay [14]. Recently B. Kleiner and B. Leeb also used asymptotic cones to prove the rigidity of quasi-isometries of symmetric spaces (see [20]).

In some cases the asymptotic cone of Γ\X is easy to describe. For example if the rank of X is 1 (or, more generally, if V is a Riemannian manifold with finite volume and strictly negative sectional curvature), then Cone(V) is a “cone” over k points, i.e., k rays with a common origin, where k is the number of ends of V . For a Riemannian product V = V_{1} × V_{2} , where V_{1} , V_{2} are as in the previous example and each has only one end, Cone(V) can be identified with the first quadrant in ℝ^{2} . Much more intricate are quotients of SL(n, ℝ)/SO(n) by congruence subgroups of SL_{n}(ℤ). These are examples of higher rank locally symmetric spaces which are not only products of rank 1 factors; their asymptotic geometry was first studied by T. Hattori (see [15], [16]). In what follows we are concerned with the general higher rank case.

By the Cartan-Hadamard theorem a simply connected, complete Riemannian manifold X of nonpositive sectional curvature is diffeomorphic to ℝ^{n}, n = dim X. One can compactify X by adjoining the boundary at infinity, ∂_{∞}X ≅ S^{n−1} , which is defined as the set of equivalence classes of asymptotic geodesic rays. The enlarged space X ⊔ ∂_{∞}X is homeomorphic to a closed ball in ℝ^{n}. On ∂_{∞}X there is a natural distance function, the Tits metric, which is also defined in terms of geodesic rays (see Section 3). In the special case where X is a symmetric space of noncompact type the boundary at infinity ∂_{∞}X carries the additional structure of a spherical Tits bulding given by the disjoint union of Weyl chambers and their walls “at infinity”. In that case the building structure on ∂_{∞}X can be reconstructed using the Tits metric (see [3]).

Since the algebraic group G is defined over ℚ one also has the Tits building T of G over ℚ, which is a simplical complex whose combinatorial structure reflects the incidence relations among parabolic ℚ–subgroups of G. The lattice Γ acts naturally on a geometric realization |T | of T as a spherical simplicial complex. The resulting quotient Γ\|T | turns out to be a finite simplical complex. If rank_{ℚ}Γ ≥ 2 the spherical complex |T | is equipped with a spherical distance function which we normalize such that the diameter of |T | is π (see Section 6). This metric induces a distance function d_{T} on the quotient space Γ\|T |. The Euclidean cone C(Γ\|T |) over Γ\|T | is the product [0, ∞) × Γ\|T | with {0} × Γ\|T | collapsed to a point and endowed with the cone metric

d_{C}^{2}((a, x), (b, y)) := a^{2} + b^{2} − 2ab cos d_{T} (x, y).

We remark that Γ\|T | is connected if ℚ–rank of Γ is ≥ 2. If rank_{ℚ}Γ = 1, |T | consists of countably many points and C(Γ\|T |) is a cone over a finite number of points as in the first example described above. In that case we set d_{C}((a, x), (b, y)) := a + b if x ≠ y and d_{C}((a, x), (b, y)) := |a − b| if x = y.

We can now describe the asymptotic cone of an arithmetic quotient:

Theorem C. Let X be a symmetric space of noncompact type (without Euclidean factor) and rank ≥ 2. Let Γ be an irreducible, non-uniform lattice in the group of isometries of X. Then the asymptotic cone of the locally symmetric space V = Γ\X is quasi-isometric to V and is isometric to the Euclidean cone over the finite simplicial complex Γ\|T |.

Various invariants of the algebraic group G are reflected in the geometry of X = G(ℝ)^{0}/K and of Γ\X. For instance it is well-known that the ℝ–rank of G equals the rank of the symmetric space X, i.e., the maximal dimension of a totally geodesic, flat submanifold in X.

It is an elementary but crucial fact that bi-lipschitz invariants of the asymptotic cone of V = Γ\X are quasi-isometry invariants of V . Theorem C thus yields the following geometric interpretation of the (purely algebraic) notion of “ℚ–rank” as a quasi-isometry invariant of V . Note that it follows from our assumptions that the ℚ–structure of G is unique (see [28]) and we can thus set rank_{ℚ}Γ := rank_{ℚ}G.

Corollary D. Let Γ be an arithmetic lattice as in Theorem C. Then

rank_{ℚ}Γ = dim Cone(Γ\X).

The dimension of the Euclidean cone C(\|T |) is defined as the (topological) dimension of the simplicial complex Γ\|T | plus one. According to [20] the asymptotic cones of a globally symmetric space X are Euclidean buildings and one has dim Cone(X) = rank_{ℝ}G.

The asymptotic cone of a compact metric space is just a point. Corollary D thus reflects geometrically a classical result of A. Borel and Harish-Chandra asserting that an arithmetic lattice Γ is co-compact, i.e., Γ\X is compact, if and only if the ℚ–rank of Γ (or G) is 0 (see [4]).

Typical for Gromov-Hausdorff limits are collapsing phenomena (as for instance the just mentioned fact that the asymptotic cone of a compact space is a point). This makes the limit objects in some sense easier to understand. On the other hand Gromov-Hausdorff limits of manifolds usually degenerate into (singular) metric spaces. In particular concepts which are related to the differentiable structure, like sectional curvature, may have no meaning in the limit. It is therefore necessary to work in a more general setting using notions that are defined for (geodesic) metric spaces. “Nonpositive sectional curvature”, for instance, can be generalized by the local property “curvature bounded from above by 0” (resp. by its global counterpart, “CAT(0) space”). Roughly speaking one requires that sufficiently small (resp. all) geodesic triangles are “thinner” than their comparison triangles of the same side length in the Euclidean plane. For example a locally symmetric space of non compact type has curvature bounded above by 0 and every globally symmetric space is CAT(0). Recent references for these purely metric concepts due to A.D. Alexandrov are [2] and the comprehensive [9].

Notice that V = Γ\X has curvature bounded above by 0, but that V (in contrast to X) is not a CAT(0) space since V is in general not contractible. However we have the following corollary to Theorem C.

Corollary E. The asymptotic cone of a locally symmetric space Γ\X (as in Theorem C) is a CAT(0) space.

The boundary at infinity ∂_{∞}V of V = Γ\X is defined as the set of equivalence classes of asymptotic geodesic rays in V . From Theorem C we deduce the following combinatorial description of the boundary at infinity of a locally symmetric space, which was first proved by L. Ji and R. MacPherson in [19] (extending methods first developed in [16] and [22]):

Theorem F. Let V = Γ\X be as in Theorem C. The points of the boundary at infinity ∂_{∞}V correspond bijectively to the points of Γ\|T |, i.e.,

∂_{∞}V ≅ Γ\|T | ≅ ∂_{∞}C(Γ\|T |).

As a consequence the distance d_{T} on Γ\|T | can be defined intrinsically in terms of geodesic rays in V (see also [19], Remark 11.9).

Corollary G. Let rank_{ℚ}Γ ≥ 2. For any two points z_{1}, z_{2} in the boundary at infinity ∂_{∞}V ≅ Γ\|T | represented by unit speed geodesic rays c_{1}, c_{2} in V , respectively, one has the formula

2 sin \frac{1}{2}d_{T} (z_{1}, z_{2}) = lim_{t→∞} \frac{1}{t}d_{V} (c_{1}(t), c_{2}(t)).

Observe that this is exactly the same formula as the one for simply connected, complete Riemannian manifolds of nonpositive curvature or more generally for CAT(0) spaces (see [3] Ch. 4, [9] and Section 5 below). For symmetric spaces the (crucial) difference is that in the simply connected case d_{T} refers to the Tits distance on the spherical building associated to G(ℝ), whereas in the non simply connected case d_{T} is induced by the Tits distance of the spherical building associated to G(ℚ). While the former (being independent of any ℚ–structure on the isometry group of X) reflects geometric properties of X, the latter depends on the ℚ–structure associated to the lattice Γ and encodes (large scale) geometric properties of the quotient V = Γ\X.

2. Coarse fundamental domains for arithmetic groups

Let G denote the identity component of the group of isometries of the symmetric space X; it is a connected, semisimple Lie group with trivial center. In this section we discuss some classical results of reduction theory about coarse fundamental domains for arithmetic groups, which are adopted to the group structure of G.

We make two natural assumptions about the non-uniform lattice Γ in G. Firstly, we assume that Γ is irreducible and, secondly, that Γ is neat and hence in particular torsion-free (see [4] §17). Under these assumptions the arithmeticity theorem of Margulis asserts that there is a connected semisimple linear algebraic group G defined over ℚ, ℚ–embedded in a general linear group GL(n, ℂ), and a Lie group isomorphism

ρ : G ⟶ G(ℝ)^{0} = (G ∩ GL(n, ℝ))^{0}

such that ρ(Γ) is arithmetic. This means that

ρ(Γ) ⊂ G(ℚ) = G ∩ GL(n, ℚ) ⊂ GL(n, ℂ)

is commensurable with the group G(ℤ) = G ∩ GL(n, ℤ) (see [30] 3.1.6 and 6.1.10). Note that the symmetric space X can be recovered as the manifold of maximal compact subgroups of the identity component of the group G(ℝ) of ℝ–rational points of G. For simplicity we identify G with G(ℝ)^{0} and Γ with ρ(Γ) in this paper.

Remark 2.1. For an irreducible, non-uniform lattice Γ as above the connected semisimple ℚ–group G is unique up to isomorphism (see [28] Section 3.2). We can thus define rank_{ℚ}Γ := rank_{ℚ}G.

2.1. Horocyclic coordinates and ℚ–Weyl chambers. We recall some definitions and basic facts about linear algebraic groups together with geometric interpretations which are needed below.

Let S (resp. T) be a maximal ℚ–split (resp. a maximal ℝ–split) algebraic torus of G, i.e. a subgroup of G which is isomorphic over ℚ (resp. ℝ) to the direct product of q (resp. r ≥ q) copies of ℂ^{∗}. All such tori are conjugate under G(ℚ) (resp. G(ℝ)) and their common dimension q (resp. r) is called the ℚ–rank (resp. ℝ–rank) of G. The identity component of S(ℝ) (resp. T(ℝ)) will be denoted by A (resp. by A_{0} ), the corresponding Lie algebra by 𝔞 (resp. by 𝔞_{0} ). The ℝ–rank of G coincides with the rank of the symmetric space X, i.e., the maximal dimension of totally geodesic flat subspaces. The choice of a maximal compact subgroup K of G is equivalent to the choice of a base point x_{0} of X. We can further choose K with Lie algebra 𝔨 so that under the corresponding Cartan decomposition 𝔤 = 𝔨 ⊕ 𝔭 of the Lie algebra 𝔤 of G we have 𝔞 ⊆ 𝔞_{0} ⊂ 𝔭 ≅ T_{x_{0}}X. Here 𝔞_{0} is maximal abelian in 𝔭, i.e., the tangent space at x_{0} of the (maximal ℝ–) flat A_{0} · x_{0} in X. The pair of Lie algebras (𝔤, 𝔞_{0}) gives rise to the root system _{ℝ}Φ of the symmetric space. Similarly there is a system of ℚ–roots _{ℚ}Φ associated to the pair (𝔤, 𝔞) (see [6] §21). It is always possible to choose orderings of _{ℚ}Φ and _{ℝ}Φ such that the restrictions of simple ℝ–roots of _{ℝ}Φ to 𝔞 are either simple ℚ–roots of _{ℚ}Φ, i.e., the elements of a basis Δ = _{ℚ}Δ of _{ℚ}Φ, or zero (see [8] 6.8). The basis _{ℝ}Δ defines a closed ℝ–Weyl chamber \overline{𝔞_{0}^{+}} in 𝔞_{0} and Δ then determines a closed ℚ–Weyl chamber in 𝔞

\overline{𝔞^{+}} := \{H ∈ 𝔞 | α(H) ≥ 0, for all α ∈ Δ\}.

We set \overline{A^{+}} = exp \overline{𝔞^{+}} (resp. \overline{A_{0}^{+}} = exp \overline{𝔞_{0}^{+}} ). A ℚ–Weyl chamber in X is a translate of the basic chamber \overline{A^{+}} · x_{0} ⊆ \overline{A_{0}^{+}} · x_{0} under G(ℚ). The elements of Δ are differentials of characters (defined over ℚ) of the maximal ℚ–split torus S. It is convenient to identify the elements of Δ also with such characters. When restricted to A we denote their values by α(a) (a ∈ A, α ∈ Δ). Notice that \overline{A^{+}} = \{a ∈ A | α(a) ≥ 1 for all α ∈ Δ\}.

A closed subgroup P of G defined over ℚ is a parabolic ℚ–subgroup if G/P is a projective variety (see [6] §11). A parabolic ℚ–subgroup P of G = G(ℝ)^{0} is by definition the intersection of G with a parabolic ℚ–subgroup of G (see [7]). The conjugacy classes under G(ℚ) of parabolic ℚ–subgroups are in one-to-one correspondence with the subsets Θ of the (chosen) set Δ of simple ℚ–roots; they are represented by the standard parabolic ℚ–subgroups P_{Θ} of G (see [6] 21.11). The corresponding standard parabolic ℚ–subgroups of G are denoted by P_{Θ} . Let P be the collection of all parabolic subgroups of G defined over ℚ. For every P ∈ P the corresponding parabolic ℚ–subgroup P = G ∩ P of G has a Langlands-decomposition P = U_{P}M_{P}A_{P} , where U_{P} is unipotent and M_{P} is reductive. Furthermore A_{P} centralizes M_{P} and normalizes U_{P} (see [7] and [4]). This yields a (generalized) Iwasawa decomposition for G, i.e., G = P K = U_{P}M_{P}A_{P}K, which in turn implies that P acts transitively on the symmetric space X. The intersection of the maximal compact subgroup K of G with M_{P} is maximal compact in M_{P} and the quotient Z_{P} = M_{P}/(K∩M_{P}) is in general the Riemannian product of a symmetric space of noncompact type by a flat Euclidean space. Let τ_{P} : M_{P} ⟶ Z_{P} be the natural projection. Then the “horocyclic coordinate map”

μ_{P} : Y_{P} = U_{P} × Z_{P} × A_{P} ⟶ X ; (u, τ_{P}(m), a) ⟼ uma · x_{0}

is an isomorphism of analytic manifolds (see [7]).

2.2. Reduction theory. A subset Ω ⊂ X is called a fundamental set for an arithmetic group Γ if the following two conditions are satified: (i) X = Γ · Ω; (ii) for every q ∈ G(ℚ) the set {γ ∈ Γ | qΩ ∩ γΩ ≠ Ø} is finite. It is the goal of reduction theory to provide explicit fundamental sets.

Let P be the minimal parabolic ℚ–subgroup of G which corresponds to the ℚ–Weyl chamber \overline{A^{+}} · x_{0} . A generalized Siegel set S = S_{ω,τ} in X (relative to \overline{A^{+}} · x_{0} ) is a subset of X of the form S_{ω,τ} = ωA_{τ} · x_{0} where ω is a relatively compact in U_{P}M_{P} and, for τ > 0, A_{τ} = {a ∈ A = A_{P} | α(a) ≥ τ, α ∈ Δ}. If we define a_{0} ∈ A by α(a_{0}) = τ for all α ∈ Δ, then A_{τ} = A_{1}a_{0} = \overline{A^{+}}a_{0} and C = A_{τ} · x_{0} ⊂ S is a (translate of a) ℚ–Weyl chamber in X. Fundamental sets for arithmetic groups consist of finitely many Siegel sets (see [4] §13 and §15 for the proofs):

Proposition 2.2. (Borel, Harish-Chandra) Let G be a semisimple algebraic group defined over ℚ with associated Riemannian symmetric space X = G/K (of maximal compact subgroups of G). Let P be a minimal parabolic ℚ–subgroup of G and let Γ be an arithmetic subgroup of G(ℚ). Then there exists a generalized Siegel set S = S_{ω,τ} such that, for a fixed set {q_{i} | 1 ≤ i ≤ m} of representatives of the finite set of double cosets Γ\G(ℚ)/P(ℚ), the union Ω = ⋃_{i=1}^{m} q_{i} · S is a fundamental set of finite volume for Γ in X.

3. Polyhedral exhaustions of locally symmetric spaces

A crucial ingredient in the proof of the main (technical) Theorem 4.1 below is the property that a locally symmetric space V = Γ\X can be exhausted (in many ways) by polyhedra, i.e., by compact submanifolds V (s) of V with corners: V = ⋃_{s≥0} V (s). Such exhaustions were constructed in [23] and further investigated in [24]. Similar constructions appear also in papers by J. Arthur, M.S. Osborne-G. Warner and L. Saper (see [1], [25] and [26]).

In this section we recall some facts about the geometric structure of the polyhedra V (s) for s ≥ 0. We invite the reader to consult the articles [23] and [24] for further details and complete proofs.

3.1. Exhaustions and the rational Tits building. Let d_{X} be the distance function induced by the Riemannian metric of the symmetric space X. A unit-speed geodesic ray is a curve c : [0, ∞) → X which realizes the distance between any two of its points, i.e., c is an isometric embedding of [0, ∞) ⊂ ℝ into X. We call two unit-speed geodesic rays c_{1}, c_{2} : [0, ∞) → X in X asymptotic if there is a constant C ≥ 0 such that lim_{t→∞} d_{X}(c_{1}(t), c_{2}(t)) ≤ C; the corresponding equivalence class is denoted by c_{1}(∞)(= c_{2}(∞)). The boundary of X at infinity, ∂_{∞}X, is defined as the set of equivalence classes of asymptotic geodesic rays. A Busemann function h associated to a ray c : [0, ∞) → X representing the equivalence class z = c(∞) ∈ ∂_{∞}X is defined by

h : X → ℝ ; x ⟼ lim_{t→∞}[d_{X}(c(t), x) − t].

A horosphere (resp. a horoball) centered at z ∈ ∂_{∞}X is a level set (resp. sublevel set) of h.

We next recall the definition of the (rational) Tits building T of G. (see e.g. [29] 5.2). Let P_{m} be the set of all maximal parabolic ℚ–subgroups of G (and ≠ G) (see Section 2). The Tits building T associated to G over ℚ is the simplicial complex whose set of vertices is P_{m}, and whose simplices are the non-empty subsets I of P_{m} such that P_{I} = ⋂_{P∈I} P is a parabolic ℚ–subgroup of G. A geometric realization |T | of T in ∂_{∞}X is obtained as follows. We define a (closed) ℚ–Weyl chamber at infinity as

\begin{aligned}(\overline{A^{+}}x_{0})(∞) & := \{c_{H}(∞) | c_{H}(t) = exp tH · x_{0}, H ∈ \overline{𝔞^{+}}, ∥H∥ = 1\} ≅ \\ & ≅ \{H ∈ \overline{𝔞^{+}} | ∥H∥ = 1\}. \end{aligned}

Any other ℚ–Weyl chamber at infinity is of the form g · (\overline{A^{+}}x_{0})(∞) with g ∈ G(ℚ). The geometric realization |T | is then defined as the union of all ℚ–Weyl chambers and their walls in ∂_{∞}X partially ordered with the order relation “wall of” (see also [19]).

The following proposition is proved in [24], Theorem 3.6.

Proposition 3.1. Let V = ⋃_{s≥0} V (s) be any (fixed) polyhedral exhaustion as introduced above. Any polyhedron V (s) can be written as Γ\X(s), where X(s) is the complement of a Γ–invariant union of countably many open horoballs:

X(s) = X − ⋃_{k} B_{k}(s).

The “centers” in ∂_{∞}X of the deleted horoballs B_{k}(s) are in one-to-one correspondence with the maximal parabolic subgroups of G defined over ℚ, or, equivalently, with the vertices of a geometric realization of the Tits building T of G.

If the ℚ–rank of Γ is 1, then the deleted horoballs B_{k}(s) are disjoint in X. If the ℚ–rank of Γ is ≥ 2, then, in contrast, the horoballs B_{k}(s) intersect and give rise to the corners of X(s) (and V (s)). Their local structure can be described for example by taking the (inner) unit normals of the involved horospheres ∂B_{k}(s) (see also [23] Lemma 4.1 (iv)). In fact, any of these horospheres is of the form γ{τ_{α}^{−1}h̃_{iα} = −s} with γ ∈ Γ, i ∈ {1, …, m}, α ∈ Δ and where the h̃_{iα} are Busemann functions on X associated to distinguished geodesic rays in the fundamental set Ω ⊂ X (see [23], Theorem 3.6).

The inner unit normal field of the horosphere {τ_{α}^{−1}h̃_{iα} = −s} in X is given by −grad h̃_{iα}. Let p be a point of V (s). The outer angle O(p) at p is defined as the set of all unit tangent vectors v ∈ T_{p}V (s) such that ⟨v, w⟩_{p} ≤ 0 for all w in the tangent cone of V (s) at p. Let Θ be a subset of Δ. If the horospheres {τ_{α}^{−1}h̃_{iα} = −s}, for α ∈ Θ, intersect, then the outer angle O(p) at an intersection point p can be identified with all positive linear combinations (of norm 1) of the unit vectors −grad h̃_{iα}(p), α ∈ Θ. From that description one can in particular see that each outer cone is isomorphic (as a cone in Euclidean space) to the face \overline{𝔞^{+}}(Θ) := \{H ∈ \overline{𝔞^{+}} | β(H) = 0, ∀ β ∈_{ℚ} Δ−Θ\} in the “model chamber” \overline{𝔞^{+}} ⊂ 𝔤 resp. \overline{A^{+}}x_{0} ⊂ X.

We define also the outer cone CO(p) at p ∈ ∂V (s) as the Euclidean cone in the tangent space T_{p}V whose vertex is 0 and whose basis is the (closed) spherical simplex O(p).

3.2. Outer angles and structure of V at infinity. The outer angles O(p) moreover reflect the structure at infinity of V = Γ\X. More precisely, we shall see below how they correspond to the spherical simplices in a geometric realization of the quotient modulo Γ of the Tits building T over ℚ associated to G.

In order to describe that quotient more explicitly we first note that the discrete, arithmetic group Γ acts on |T |. The resulting quotient space Γ\|T | turns out to be a finite simplicial complex. In fact, as mentioned in Section 2, the conjugacy classes of elements of the collection P of parabolic ℚ–subgroups of G are in one-to-one correspondence with the subsets Θ of the set _{ℚ}Δ of simple ℚ–roots. Every conjugacy class has a standard representative denoted by P_{Θ} . One can show that the sets of double cosets Γ\G(ℚ)/P_{Θ}(ℚ) are finite for all Θ (see [4] 15.6). Let △ ⊂ ℝ^{q} be the spherical q − 1 simplex obtained by taking unit vectors in \overline{𝔞^{+}} ⊂ 𝔞 ≅ ℝ_{q} (q = ℚ–rank of G); note that with our notation above △ ≅ (\overline{A^{+}}x_{0})(∞). For a subset Θ of _{ℚ}Δ we define the boundary simplex △(Θ) of △ as △(Θ) := △ ∩ {α = 0 | α ∈_{ℚ} Δ − Θ}. A vertex of △ then corresponds to a maximal parabolic ℚ–subgroup while the entire simplex △ corresponds to the minimal parabolic ℚ–subgroup P := P_{Ø} of G. Let the set Γ\G(ℚ)/P(ℚ) be represented by {q_{1}, … q_{m}} (see Proposition 2.2) and take m isometric copies △^{j} of △ with faces △^{j}(Θ) corresponding to Θ. The corresponding isometries △ ≃ △^{j} are denoted by φ_{j}. The simplicial complex Γ\|T |, which provides a geometric realization of the quotient of the Tits building |T | modulo Γ, is constructed from the simplices △^{1}, …, △^{m} through the following incidence relations:

Two simplices △^{j} and △^{l} are pasted together along the faces △^{j}(Θ) and △^{l}(Θ) by the isometry φ_{j}∘φ_{l}^{−1} |_{△^{l}(Θ)} if and only if Γq_{j}P_{Θ}(ℚ) = Γq_{l}P_{Θ}(ℚ). The simplicial complex |T | and hence also Γ\|T | is connected if the ℚ–rank q is greater than or equal to two; moreover dim Γ\|T | = rank_{ℚ} Γ − 1.

Having constructed the finite simplical complex Γ\|T | we briefly explain how its simplices correspond to outer angles of the polyhedra V (s) for an exhaustion V = ⋃_{s≥0} V (s) of V . Let E be the set of double cosets Γ\G(ℚ)/P_{Θ}(ℚ), Θ ⊂_{ℚ} Δ. By [4] 15.6, E is a finite set.

By definition the elements of E are in one to one correspondence to the Γ–equivalence classes of parabolic ℚ–subgroups of G or the simplices of Γ\|T | (see the above construction or [24], Section 4). The elements of E also index the boundary strata of V (s). In fact, if p is some point in the interior of a boundary stratum of V (s), then there is a unique set Θ of simple roots such that the outer angle at p is spanned by the gradients of the intersecting horospheres −grad h̃_{iα}, α ∈ Θ, and hence O(p) ≅ △(Θ) (and CO(p) ≅ \overline{𝔞^{+}}(Θ)).

In [24], Section 4, we showed that every point in the complement of the interior V (s)^{o} of V (s) in V is contained in the image under the Riemannian exponential map of the outer cone CO(p) at some point of ∂V (s). More precisely we have

Proposition 3.2. Let V = ⋃_{s≥0} V (s) be any (fixed) polyhedral exhaustion as described above. Then there is a disjoint union

V − V (s)^{0} = ∐_{p∈∂V (s)} Exp_{p}CO(p).

Moreover, the points at infinity of those unit speed geodesic rays whose initial vectors are in O(p) span a (closed) spherical simplex σ_{E}, E ∈ E, of the finite complex Γ\|T |.

3.3. The fine structure of the boundary strata. We will need some further details about the structure of the boundary strata of a polyhedral exhaustion. For proofs we refer the reader to [24], Section 4. Pick E = E(Θ) ∈ E and let σ_{E} be the corresponding simplex of the finite complex Γ\|T |. The pointwise isotropy group of σ_{E} is isomorphic to the group of real points, say P_{E}, of a conjugate under Γ of the standard parabolic subgroup P_{E} of G (see [23], Lemma 1.2). Recall from Section 2 that there is a Langlands decomposition P_{E} = U_{E}M_{E}A_{E} and an associated decomposition X ≅ U_{E} × Z_{E} × A_{E}. The arithmetic group Γ ∩ P_{E} has trivial A_{E} factor and has a finite index subgroup which is the semidirect product of Γ ∩ U_{E} with Γ ∩ M_{E}. A truncated locally symmetric space Γ ∩ M_{E}\Z_{E}(s) is a compact submanifold with corners in a totally geodesic subspace of V isometric to Γ ∩ M_{E}\Z_{E}.

Proposition 3.3. Let V = ⋃_{s≥0} V (s) be a polyhedral exhaustion of V . Then, for each s, the boundary ∂V (s) consists of a finite number of strata V_{E}(s), E = E(Θ) ∈ E, each of which is a fibre bundle over a truncated locally symmetric space with a compact nilmanifold as fibre:

0 → Γ ∩ U_{E}\U_{E} → V_{E}(s) → Γ ∩ M_{E}\Z_{E}(s) → 0.

The combinatorics of these strata is “dual” to that of the simplicial complex Γ\|T |, i.e., (n − k)-dimensional strata of ∂V (s) correspond to (k − 1)-simplices of Γ\|T |.

Moreover, the outer cones CO(p) at the interior points of V_{E}(s) are all isomorphic to some face \overline{𝔞^{+}}(Θ) (uniquely determined by the index E = E(Θ)) of the ℚ–Weyl chamber \overline{𝔞^{+}} .

We now consider the universal covering space X(0) ⊂ X of the polyhedron V (0) of some exhaustion of V . The distance function d_{V} on V is defined by d_{V} (π(x), π(y)) = inf_{γ∈Γ} d_{X}(x, γ · y), where π : X → V = Γ\X denotes the canonical projection. The next lemma will be used in the proof of Theorem 4.1.

Lemma 3.4. Let CO(z) ⊂ X be the outer cone of the submanifold with corners X(0) ⊂ X. For x ∈ CO(z) the geodesic segment [π(x)π(z)] realizes the shortest distance (in V ) from π(x) to the compact manifold with corners V (0) = Γ\X(0); and, in particular, d_{V} (π(x), π(y)) = d_{X}(x, z).

Proof. Assume that there is a point π(y) ∈ V (0) such that

d_{V} (π(x), π(y)) < d_{V} (π(x), π(z)) = d_{X}(x, z);

where the equality comes from the fact (see [24], Corollary 3.5.) that π(x) lies on a ray in V starting at π(z). Since d_{V} (π(x), π(y)) = inf_{γ∈Γ} d_{X}(x, γy) the assumption implies that for some γ ∈ Γ, d_{X}(x, γy) < d_{X}(x, z). Since X(0) is Γinvariant and y ∈ X(0) we have γy ∈ X(0) and hence d_{X}(x, X(0)) < d_{X}(x, z). This is a contradiction since the segment [xz] clearly realizes the shortest distance in X from x to X(0).

We conclude this section with an interesting metric property of Γ\|T | which will be used in Section 5. The definition of a “CAT(1) space” can be found for example in [9] or [2].

Lemma 3.5. The finite spherical simplical complex (Γ\|T |, d_{T}) is a CAT(1) space.

Proof. The model simplex △ and all its faces are convex subsets of the unit sphere in ℝ^{q} and therefore CAT(1) spaces. Pasting CAT(1) spaces along complete (π)–convex subspaces yields again CAT(1) spaces (see e.g. [9], Ch.II. 4.4).

4. Metric properties of arithmetic quotients of symmetric spaces

4.1. Isometric images of Weyl chambers. In this section we derive the crucial estimates. We will prove that the canonical projection map π : X ⟶ V is an isometry (with respect to the distance function d_{X} and d_{V} ) when restricted to certain closed ℚ–Weyl chambers. The following theorem improves previous results of Hattori, Ding and Ji on metric properties of Siegel sets.

Theorem 4.1. Let V = ⋃_{s≥0} V (s) be an exhaustion of V by manifolds with corners (as in Section 3) and let CO(p), p ∈ V (0), be any outer cone of V (0). Then the restriction of the canonical projection π : X ⟶ V to any lifted outer cone CO(z) ⊂ X, with π(z) = p, is an isometry with respect to the distance functions of X and V , respectively.

In particular, there is a ℚ–Weyl chamber C_{∗} ⊂ C such that for every i ∈ {1, …, m} the chamber q_{i}C_{∗} ⊂ Ω ⊂ X is isometrically mapped to V under π.

Proof. We consider an arbitrary exhaustion V = ⋃_{s≥0} V (s). For a point π(z) ∈ V (0) let CO(π(z)) be the outer cone. In the lift of CO(z) of CO(π((z)) to X we pick two arbitrary points x, y. We wish to show that d_{V} (π(x), π(y)) ≥ d_{X}(x, y) (the opposite inequality holds by definition). Let c be a curve of minimal length in V between π(x) and π(y), i.e., L(c) = d_{V} (π(x), π(y)). We distinguish two cases: (1) c intersects the compact polyhedron V (0) and (2) c does not intersect V (0).

Case (1): Decompose c into three segments c = c_{1} ∪ c_{2} ∪ c_{3} , such that c_{1} connects π(x) to the first intersection point of c with V (0) and c_{3} connects the last intersection point of c with V (0) to π(y). By Lemma 3.4. we have

d_{V} (π(x), V (0)) = d_{V} (π(x), π(z)) = d_{X}(x, z) and
d_{V} (π(y), V (0)) = d_{V} (π(y), π(z)) = d_{X}(y, z).

Hence

d_{V} (π(x), π(y)) = L(c) = L(c_{1}) + L(c_{2}) + L(c_{3}) ≥ d_{X}(x, z) + d_{X}(y, z) ≥ d_{X}(x, y).

Case (2): According to Propositions 3.2 and 3.3 we can decompose the curve

c : [0, d_{V} (π(x), π(y))] ⟶ V \ V (0)

into a finite number of segments c_{i}, 1 ≤ i ≤ l, say, each of which is contained in a subset of V diffeomorphic to W_{E}(0) × \overline{𝔞^{+}}(Θ), where W_{E}(0) is a nilmanifold bundle over truncated locally symmetric and compact and the ℚ-Weyl chamber face \overline{𝔞^{+}}(Θ) ≅ \overline{A^{+}}(Θ)x_{0} corresponds to the (congruent) outer cones with E = E(Θ). The Riemannian metric on X with respect to horocyclic coordinates corresponding to A_{E} supporting \overline{A^{+}}(Θ) is given by a formula of Borel (see [5], Proposition 1.6). From the Γ-invariance of the exhaustion in X (compare the proof of Theorem 4.2 in [23]) we deduce that the Riemannian metric on V at the point (w, a) ∈ W_{E}(0) × \overline{A^{+}}(Θ) can be written in the form

ds_{(w,a)}^{2} = dw_{(w,a)}^{2} + da_{a}^{2}.

For the length of the segment c_{i}(s) = (w_{i}(s), a_{i}(s)), s ∈ [s_{i}, s_{i+1}], we thus have the estimate L(c_{i}) ≥ L(a_{i}), 1 ≤ i ≤ l. Hence

d_{V} (π(x), π(y)) = L(c) = ∑_{i=1}^{l} L(c_{i}) ≥ ∑_{i=1}^{l} L(a_{i}).

Now consider the continuous curve a(s) := ⋃_{i=1}^{l} a_{i}(s) in the chamber \overline{A^{+}} ⊂ A. If we write x = ax_{0} and y = bx_{0} , the curve a(s)x_{0} connects x and y in \overline{A^{+}}x_{0} and satisfies

∑_{i=1}^{l} L(a_{i}) ≥ d_{A}(a, b) = d_{X}(x, y).

Together with the estimate above this proves the claim also in case (2).

Finally, the chamber C_{∗} can be chosen as follows. Consider the preimages in the fundamental set Ω of the minimal boundary strata of ∂V (0). They intersect the ℚ–Weyl chambers q_{i}C, i = 1, …, m, in well defined points x_{i} (see [23] Lemma 3.5). The lifted (maximal) outer cones based at the points π(x_{i}) of ∂V (0) are again ℚ–Weyl chambers q_{i}C_{∗} ⊆ q_{i}C, i = 1, …, m, and in finite Hausdorff-distance from q_{i}C.

4.2. A net in V . Theorem 4.1 yields a metric space which is quasiisometric to V . Recall that a subset N of a metric space (S, d) is called a (ε–)net if there is some positive constant ε such that d(s, N) ≤ ε for all s ∈ S; in particular the Hausdorff-distance between N and S is at most ε.

Corollary 4.2. There is a net N in V consisting of finitely many isometrically embedded ℚ–Weyl chambers. Proof. Let C_{∗} be as in Theorem 4.1. The definition of a fundamental set as a finite union of Siegel sets Ω = ⋃_{i=1}^{m} q_{i} · S (see Proposition 2.2) implies that ⋃_{i=1}^{m} q_{i} · C and hence also ⋃_{i=1}^{m} q_{i} · C_{∗} is a net in Ω ⊂ X. Since V = π(Ω) and d_{V} (π(x), π(y)) ≤ d_{X}(x, y) for all x, y ∈ X the claim follows.

Remark 4.3. From Proposition 3.1 (see also [23], Lemma 2.4, 2.5) it follows that the q_{i}C_{∗}, 1 ≤ i ≤ m, can be chosen to be disjoint in X and such that π : X → V restricted to ⋃_{i=1}^{m} q_{i}C_{∗} is bijective.

5. The asymptotic cone and the proof of a conjecture of Siegel

5.1. A quasi-isometric model for a locally symmetric space. The goal of this section is to show that the asymptotic cone of a non-compact locally symmetric space V = Γ\X of finite volume is isometric to the Euclidean cone over a finite simplicial complex. The latter is given as the quotient of the Tits bulding T of G modulo Γ.

If rank_{ℚ}Γ ≥ 2 the geometric realization |T | of T as a spherical complex carries a natural distance function which we normalize such that the diameter of |T | is π. This “spherical” metric induces a distance function d_{T} on the finite simplical complex Γ\|T |. We define the Euclidean cone C(Γ\|T |) over Γ\|T | to be the product [0, ∞) × Γ\|T | with {0} × Γ\|T | collapsed to a point O and endowed with the cone metric

d_{C}^{2}((a, x), (b, y)) := a^{2} + b^{2} − 2ab cos d_{T} (x, y).

If rank_{ℚ}Γ = 1 we set d_{C}((a, x), (b, y)) := a + b if x ≠ y and d_{C}((a, x), (b, y)) := |a − b| if x = y.

In the same way one defines the Euclidean cone over an arbitrary metric space of diameter ≤ π (see e.g. [9]).

There is an alternative way to build up (C(Γ\|T |), d_{C}) which parallels the construction of Γ\|T | in Section 3: We there realized the spherical simplices △^{j} for j = 1, …, m on the unit sphere in ℝ^{q}; thus we can take the cones C(△^{j}) in the Euclidean space ℝ^{q} with vertex 0. We endow these simplicial cones with the induced Euclidean metric and glue them together using the same incidence relations as for Γ\|T |.

We recall the notion of Hausdorff-convergence of (unbounded) pointed metric spaces (see [13] or [10]). The distortion of a map f : A → B of metric spaces A and B is defined as

dis(f) := sup_{a,b} |d_{A}(a, b) − d_{B}(f(a), f(b))|.

The uniform distance between metric spaces A and B is defined as |A, B|_{u} = inf_{f} dis(f) where the infimum is taken over all bijections f : A → B. A sequence of metric spaces M_{n} Hausdorff-converges to a metric space M iff for every ε > 0 there is an ε–net M_{ε} in M which is the uniform limit of ε–nets (M_{n})_{ε} in M_{n}. We say that a sequence (M_{n}, p_{n}) of unbounded, pointed metric spaces Hausdorffconverges to a pointed metric space (M, p) if for every r > 0 the balls B_{r}(p_{n}) in M_{n} Hausdorff-converge to the ball B_{r}(p) in M.

Let v_{0} be an (arbitrary) point of the locally symmetric space V = Γ\X. We define the asymptotic cone of V as the Hausdorff-limit of pointed metric spaces:

Cone(V) := H − lim_{n→∞}(V, v_{0}, \frac{1}{n}d_{V}).

By Corollary 4.2 there is a net N in V consisting of disjoint closed Euclidean cones π(q_{i}C_{∗})) in V . We identify the abelian Lie algebra 𝔞 with ℝ^{q}. For all n ∈ ℕ we define a map

f_{n} : N ⊂ (V, \frac{1}{n}d_{V}) → C(Γ\|T |); by

f_{n}(π(q_{j}(exp H) · x_{0})) := \frac{1}{n}H ∈ log C_{∗} ≅ C(△^{j}) ⊂ ℝ^{q}.

By Remark 4.3 and the second construction of C(Γ\|T |) f_{n} is a bijection from the interior of N onto its image in C(Γ\|T |); note that this image is open and dense in C(Γ\|T |) for all n.

Lemma 5.1. There is a constant D ≥ 0 such that dis(f_{n}) ≤ \frac{1}{n}D for all n ∈ ℕ, i.e., f_{n} is a (1, \frac{1}{n}D)–quasi-isometry. In particular, V is quasi-isometric to C(Γ\|T |).

Proof. We consider some polyhedral exhaustion V = ⋃_{s≥0} V (s). The intersection of the net N with V \V (0) is still a net in V since V (0) is compact. Let u, v be two points in the interior of N ∩ (V \ V (0)). We take a path c([0, L]) in V between u and v of minimal length and parametrized by arc-length. By Proposition 2.2 we can write V = π(Ω) = ⋃_{j=1}^{m} q_{j}S for a fundamental set Ω. We can thus find t_{i} ∈ [0, L], 0 ≤ i ≤ n + 1 such that t_{0} = 0, t_{n+1} = L and [0, L] = ⋃_{i=0}^{n}[t_{i}, t_{i+1}] with c([t_{i}, t_{i+1}]) ⊂ π(q_{j_{i}}S) for j_{i} ∈ {1, …, m}. Associated to the path c we thus get a well-defined string (j_{0}, j_{1}, …, j_{n}). We next replace c by a path \overline{c} whose associated string contains an element j_{k} ∈ \{1, …, m\} at most once: Start with j_{0} and assume that 0 ≤ l ≤ n is the greatest index such that j_{l} = j_{0} . Geometrically this means that the path c returns to π(q_{j_{0}}S) at c(t_{l}). Using Theorem 4.1 and the definition of a Siegel set S we can replace c([t_{0}, t_{l+1}]) by a geodesic segment, say \overline{c}([s_{j_{0}}, s_{j_{0}+1}]), in π(q_{j_{0}}C) whose length L satisfies

L(\overline{c}([s_{j_{0}}, s_{j_{0}+1}])) ≤ L(c[t_{0}, t_{l+1}]) + 2D_{1},

for D_{1} := diam(ω). Repeating this procedure with j_{l+2} etcetera we eventually get a sequence of at most m segments \overline{c}([s_{k}, s_{k+1}]) in N of total length ≤ L(c)+2mD_{1}. We next show that these segments can be chosen in such a way that they are mapped by f_{1} to a continous path c̃ in C(Γ\|T |) from f_{1}(u) to f_{1}(v).

In fact, each π(q_{j}C_{∗}) ⊂ π(q_{j}S) ⊂ V is isometric to a simplicial cone in C(Γ\|T |). The map f_{1} thus yields well defined images of (the interior of) the segments \overline{c}([s_{k}, s_{k+1}]) ∈ π(q_{k}C_{∗}). By construction the endpoint \overline{c}(s_{k−1}) ∈ π(q_{k−1}C_{∗}) and the initial point \overline{c}(s_{k}) ∈ π(q_{k}C_{∗}) are on the same levelset, say ∂V (s), of the exhaustion of V we have chosen above and by construction are at most the distance 2D_{1} apart. By the (second) construction of C(Γ\|T |) we can join their images by segments in C(Γ\|T |) of uniformly bounded length 2D_{1} to obtain the path c̃. This argument yields the estimate

d_{C}(f_{1}(u), f_{1}(v)) ≤ L_{C}(c̃) ≤ 4mD_{1} + L(c) = 4mD_{1} + d_{V} (u, v).

On the other hand, given a geodesic path c̃ in C(Γ\|T |) joining two points x and y in the image of the interior of N ∩ (V \ V (0)) under f_{1} , we can lift the segments contained in the simplicial cones C(△^{j}) to the corresponding chambers π(q_{j}C_{∗}) via f_{1}^{−1} . By the same arguments as before the endpoints of the lifted segments can be joined in V to form a continuous path c between f_{1}^{−1}(x) and f_{1}^{−1}(y) of length

(2)  d_{V} (f_{1}^{−1}(x), f_{1}^{−1}(y)) ≤ L(c) ≤ 2mD_{1} + L(c̃) ≤ 2mD_{1} + d_{C}(x, y).

Combining (1) and (2) and setting D := 4mD_{1} we get for all u, v in (the interior of) N ∩ (V \ V (0)) that

|d_{V} (u, v) − d_{C}(f_{1}(u), f_{1}(v))| ≤ D.

Finally since for any n ∈ ℕ

\begin{aligned}|\frac{1}{n}d_{V} (u, v) − d_{C}(f_{n}(u), f_{n}(v))| & = |\frac{1}{n}d_{V} (u, v) − \frac{1}{n}d_{C}(f_{1}(u), f_{1}(v))| = \\ & = \frac{1}{n}|d_{V} (u, v) − d_{C}(f_{1}(u), f_{1}(v))| ≤ \frac{1}{n}D, \end{aligned}

the Lemma follows.

We are now prepared to state a key result of this paper. The following theorem has been proved independently by T. Hattori. His approach is not intrinsic and uses equivariant Hausdorff limits with respect to suitable finite index subgroups of Γ and an embedding into a space of positive definite matrices (see [17]).

Theorem 5.2. Let X be a Riemannian symmetric space of rank ≥ 2, let Γ be an irreducible, non-uniform lattice in the group of isometries of X and let V be the locally symmetric space Γ\X. Then the asymptotic cone Cone(V) is isometric to the Euclidean cone C(Γ\|T |) over the finite simplicial complex Γ\|T |. Proof. Given ε > 0 there is n_{ε} ∈ ℕ, such that N′ := N ∩(V \V (0)) ⊂ V is an ε–net in (V, \frac{1}{n}d_{V}) for all n ≥ n_{ε}. Since f_{1}(N′) is dense in C(Γ\|T |)\f_{1}(N∩V (0)) it is an ε–net in the latter for all ε > 0. Then there is r_{0} such that for r ≥ r_{0} the same assertions are true for the subsets of balls with radius r:

N′ ∩ B_{r}(v_{0}) ⊂ (V, \frac{1}{n}d_{V}) and f_{n}(N′) ∩ B_{r}(O) ⊂ C(Γ\|T |).

From Lemma 5.1 and its proof we obtain the following uniform estimate:

|N′ ∩ B_{r}(v_{0}), f_{n}(N′) ∩ B_{r}(O)|_{u} ≤ \frac{1}{n}D,

which implies the theorem.

The definition of the asymptotic cone of a metric space S implies that bi-lipschitz invariants of Cone(S) are quasi-isometry invariants of S (see [14] for a discussion). It is well-known that the ℝ–rank of G equals the rank of the symmetric space X, i.e., the dimension of a maximal totally geodesic submanifold of X with sectional curvature zero. The dimension of a simplicial complex is equal to the dimension of a maximal simplex. We thus define

dim Cone(Γ\X) := dim(Γ\|T |) + 1.

Theorem 5.2 yields the following geometric interpretation of the ℚ–rank and the cohomological dimension, cd Γ, of Γ:

Corollary 5.3. Let X be a symmetric space of noncompact type and rank ≥ 2 and let Γ be an irreducible (arithmetic) lattice in the isometry group of X. Then

(i)  rank_{ℚ}Γ = dim Cone(Γ\X)
(ii)  cd Γ = dim X − dim Cone(Γ\X).

Proof. By a well-known result of Borel and Harish-Chandra V is compact if and only if rank_{ℚ}Γ = 0 (see [4]). If V = Γ\X is compact then Cone(V) is a point, i.e., dim Cone(V) = 0. On the other hand if V is not compact there is a geodesic ray in V and thus dim Cone(V) ≥ 1. This proves the formula in the compact case. If V is not compact, i.e., if rank_{ℚ}Γ ≥ 1, we have by Theorem 5.2

dim Cone(Γ\X) = dim(Γ\|T |) + 1 = dim |T | + 1 = rank_{ℚ}G = rank_{ℚ}Γ

and hence (i). Equality (ii) follows from (i) and [7] 11.4.3.

Remark 5.4. In [20] B. Kleiner and B. Leeb showed that the asymptotic cone (with respect to any fixed ultrafilter) of a globally symmetric space X is a Euclidean building. The rank of this building is equal to the rank of X, i.e., to the ℝ–rank of G. The latter is therefore a quasi-isometry invariant of X, whereas the ℚ–rank is a quasi-isometry invariant of Γ\X.

Corollary 5.5. The asymptotic cone of a locally symmetric space V = Γ\X is a CAT(0) space and thus in particular contractible.

Proof. By Theorem 5.2 Cone(V) is isometric to the Euclidean cone C(Γ\|T |). A theorem of V.N. Berestovski says that for a geodesic metric space Y the Euclidean cone C(Y) is a CAT(0) space iff Y is a CAT(1) space (see [9], Ch. II.4). The claim then follows from Lemma 3.4.

q_{j}x 3 2 1 = 1′ q_{i}y 2′ 3′ q_{i}x Figure 5.1

5.2. A conjecture of Siegel and its proof.

Remark 5.6. Theorem 4.1 could be deduced from an estimate about the distance function d_{X} of X when restricted to a fundamental set Ω, which is given in a paper by A. Borel (see [5], Theorem 2.3). But unfortunately the second part of the proof given there only works in the ℚ–rank 1 case and is not complete for ℚ–rank ≥ 2 (this was also observed in the preprint version of [19]). More precisely, in [5], pp. 550-552, inequality (12) does not imply inequality (14) and this in turn does not imply inequality (5) as is claimed there. Nevertheless the assertion of the theorem, which in particular answers a question of C.L. Siegel (see [27], Section 10), is correct and we next present a complete proof based on Theorem 4.1 and Lemma 5.1.

Theorem 5.7. For q_{k} ∈ G(ℚ) (k = 1, …, m) and a Siegel set S as in Proposition 2.2 there is a constant D′ such that

d_{X}(γq_{i}x, q_{j}y) ≥ d_{X}(x, y) + D′,

for all i, j ∈ {1, …, m}, all γ ∈ Γ and all x, y ∈ S. In particular the canonical projection π : X ⟶ Γ\X restricted to each Siegel set q_{i}S is a (1, D)– quasiisometry.

Proof. The definition of the Siegel set S implies that the ℚ–Weyl chamber C_{∗} is a net in S. Hence it suffices to prove the estimate for x, y ∈ C_{∗}. By definition of d_{V} and Lemma 5.1 we have

\begin{aligned}d_{X}(γq_{i}x, q_{j}y) & ≥ d_{V} (π(q_{i}x), π(q_{j}y)) ≥ \\ & ≥ d_{C}(f_{1}(π(q_{i}x)), f_{1}(π(q_{j}y))) − D. \end{aligned}

We can further decompose a shortest path c in C(Γ\|T |) into segments contained in the simplicial cones C(△^{l}) ⊂ C(Γ\|T |). Since all these Euclidean cones are isometric (for l = 1, …, m) we can take copies of successive segments to build a continuous path c′ of the same length as c in a single cone, say C(△^{i}). In Fig. 5.1 is indicated the “shadow” of c and c′ in the basis Γ\|T | of the cone C(Γ\|T |). It then follows that

\begin{aligned}d_{C}(f_{1}(π(q_{j}x)), f_{1}(π(q_{i}y))) & ≥ d_{C}(f_{1}(π(q_{i}x)), f_{1}(π(q_{i}y))) ≥ \\ & ≥ d_{V} (π(q_{i}x), π(q_{i}y)) − D, \end{aligned}

where for the last inequality we used Lemma 5.1 again. Finally, by Theorem 4.1, we have that d_{V} (π(q_{i}x), π(q_{i}y)) = d_{X}(q_{i}x, q_{i}y) = d_{X}(x, y); this proves the theorem.

Theorem 5.7 has independently been proved by Ji and in a special case by Ding (see Section 1).

6. On the Tits geometry of locally symmetric spaces

We consider a locally symmetric space V = Γ\X as in the previous section 5, i.e., X is a symmetric space of non-compact type and rank ≥ 2 and Γ is an irreducible, non-uniform (arithmetic) lattice in the isometry group of X. We remark in passing that the boundary at infinity ∂_{∞}X of the globally symmetric space X can be identified with a geometric realization of the spherical Tits building associated to G(ℝ); its dimension is equal to rank X −1 = rank_{ℝ}G−1. The following combinatorial description of the boundary at infinity of the locally symmetric space V in terms of the Tits building T associated to G(ℚ) is due to Ji and MacPherson (see [19]). As an application of Theorem 5.2 we obtain a new proof of this result.

Theorem 6.1. The equivalence classes of asymptotic geodesic rays in the locally symmetric space V correspond bijectively to the points of Γ\|T |, i.e.,

∂_{∞}V ≅ Γ\|T |.

Before proving this theorem we describe the geodesic rays in the Euclidean cone C(Γ\|T |).

Lemma 6.2. Any geodesic ray c in C(Γ\|T |) is of the form c(t) = (t_{0} + t, z) for some t_{0} ≥ 0 and z ∈ Γ\|T |. Moreover two rays c_{1} and c_{2} are asymptotic iff there are t_{1}, t_{2} ≥ 0 with c_{1}(t) = (t_{1} + t, z) and c_{2}(t) = (t_{2} + t, z), i.e., ∂_{∞}C(Γ\|T |) ≅ Γ\|T |.

Proof. Let c : [0, ∞) → C(Γ\|T |) be a geodesic ray, i.e., a curve which realizes the distance between any two of its points. If c eventually lies in a single chamber C(△^{k}) considered as a subset of C(Γ\|T |), then it is clearly of the claimed form. We may thus assume, that the ray c does not eventually stay in a single chamber. It then has to return to a fixed chamber, say C(△^{j}), i.e., there are 0 ≤ t′ < t′′ and 1 ≤ j ≤ m such that c(t′), c(t′′) ∈ C(△^{j}). Since the simplicial cones C(△^{k}) ⊂ C(Γ\|T |), k = 1, …, m, are all isometric we can take copies of the segments of c([t′, t′′]) in different chambers to build a continuous path of the same length d_{C}(c(t′), c(t′′)) = t′′ − t′ in C(△^{j}) (compare the proof of Theorem 5.7). This is a contradiction, because by construction (and the assumption about c) this path is strictly longer than the straight line in C(△^{j}) between the points c(t′) and c(t′′) on the geodesic ray c.

Proof. (of Theorem 6.1). Let z ∈ ∂_{∞}V be represented by a unit-speed geodesic ray c : [0, ∞) → V , i.e., z = c(∞). Then c is also a geodesic ray in the rescaled space (V, \frac{1}{n}d_{V}) and hence c converges to a ray ĉ in Cone(V) = lim_{n→∞}(V, v_{0}, \frac{1}{n}d_{V}). If one also has z = c̃(∞) then the rays c̃ and c are in finite Hausdorff distance in V and thus converge to the same ray ĉ in Cone(V). By Theorem 5.2 Cone(V) is the Euclidean cone C(Γ\|T |). Using Lemma 6.2 we thus get a well defined map

R : ∂_{∞}V ⟶ Γ\|T |; c(∞) ⟼ ĉ(∞),

which we claim to be bijective.

We first show that R is onto: Let z ∈ Γ\|T |. The simplicial complex Γ\|T | was constructed by pasting together finitely many simplices △^{k}. Thus there is at least one simplex, say △^{j}, such that z ∈ △^{j} ⊂ Γ\|T |. We have a natural identification q_{j}C_{∗}(∞) ≅ △^{j} (see Section 3) and by Theorem 4.1 there is a ray of the form π(q_{j}c(t)) ⊂ π(q_{j}C_{∗}) ⊂ V with R(π(q_{j}c(∞))) = ĉ(∞) = z.

Next we show that R is one-to-one: Assume that for two rays c_{1} and c_{2} in the net N ⊂ V one has

ĉ_{1}(∞) = R(c_{1}(∞)) = z = R(c_{2}(∞)) = ĉ_{2}(∞).

By Theorem 5.2 and Lemma 6.2 the rays c_{1} (resp. c_{2} ) Hausdorff-converge to ĉ_{1}(t) = (t_{1} + t, z) (resp. ĉ_{2}(t) = (t_{2} + t, z)). This implies that there is n_{0} ∈ ℕ and some constant κ, such that for all n ≥ n_{0} and all t ∈ [0, 1] one has (see proof of Lemma 5.1)

\frac{1}{n}d_{V} (c_{1}(tn), c_{2}(tn)) ≤ \frac{κ}{n}.

Consequently the rays c_{1} and c_{2} are asymptotic in V .

Let X be a simply connected, complete Riemannian manifold of nonpositive curvature and let z_{1}, z_{2} ∈ ∂_{∞}X represented by geodesic rays c_{1} and c_{2} , respectively. The Tits metric on ∂_{∞}X is defined by the formula

2 sin \frac{1}{2}d_{T} (z_{1}, z_{2}) := lim_{t→∞} \frac{1}{t}d_{X}(c_{1}(t), c_{2}(t)).

This generalizes the corresponding formula for a globally symmetric space of non-compact type where d_{T} is the Tits distance of the spherical Tits building of G(ℝ). See the book [12] for details. The next result asserts that exactly the same intrinsic formula also holds in the case of non simply connected, arithmetic quotients.

Corollary 6.3. For two points z_{1}, z_{2} in ∂_{∞}V ≅ Γ\|T | represented by unit speed geodesic rays c_{1}(t), c_{2}(t) in V , respectively, one has the formula

2 sin \frac{1}{2}d_{T} (z_{1}, z_{2}) = lim_{t→∞} \frac{1}{t}d_{V} (c_{1}(t), c_{2}(t)).

Proof. By Theorem 5.2 and the definitions the balls of radius 1 in (V, \frac{1}{t}d_{V}) centered at v_{0} Hausdorff-converge to the ball of radius 1 centered at O in C(Γ\|T |). Hence using Theorem 6.1 we have

lim_{t→∞} \frac{1}{t}d_{V} (c_{1}(t), c_{2}(t)) = d_{C}((1, z_{1}), (1, z_{2})) =

= [2 − 2 cos d_{T} (z_{1}, z_{2})]^{1/2} = 2 sin \frac{1}{2}d_{T} (z_{1}, z_{2}).

Alternatively, one can identify the rays c_{1} , c_{2} with rays in the cone C(Γ\|T |) and then use Corollary 5.5 and the fact the formula holds in any CAT(0) space (see [9], Ch.III 3.6).

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338 Leuzinger

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Enrico Leuzinger Mathematisches Institut II Universität Karlsruhe Englerstr. 2 D-76131 Karlsruhe Germany enrico.leuzinger@math.uni-karlsruhe.de

Received March 07, 2002 and in final form November 1, 2003