MOSTOW-MARGULIS RIGIDITY WITH LOCALLY COMPACT TARGETS

ALEX FURMAN

Abstract. Let Γ be a lattice in a simple higher rank Lie group G. We describe all locally compact (not necessarily Lie) groups H in which Γ can be embedded as a lattice. For lattices Γ in rank one groups G (with the only exception of non-uniform lattices in G ≃ SL_{2}(ℝ), which are virtually free groups) we give a similar description of all possible locally compact groups H, in which Γ can be embedded as a uniform lattice.

1. Introduction and Statement of the Main Results

Throughout this paper we use the following terminology: (semi)simple Lie group stands for (semi)simple, connected real Lie group with finite center and no non-trivial compact factors. If (semi)simple Lie groups G, G′ are locally isomorphic we write G ≃ G′. Locally compact groups are assumed to be second countable, but otherwise may be very general. A countable subgroup Γ in a locally compact group G is said to form a lattice if it is discrete and G/Γ carries a finite G-invariant measure (note that any locally compact group which has a lattice is necessarily unimodular). A lattice is said to be uniform if G/Γ is compact. An embedding π : Γ⟶G of a countable group in a locally compact group G is said to be a lattice embedding (resp. uniform lattice embedding) if π(Γ) forms a lattice (resp. uniform lattice) in G. A lattice Γ in a semisimple Lie group G = ΠG_{i} is called irreducible if its projection on each of the simple factors G_{i} is dense.

The starting point of our discussion is Mostow’s rigidity:

Strong Rigidity Theorem (Mostow, Prasad, Margulis). Let G and H be semisimple Lie groups, where G ≄ SL_{2}(ℝ). Let Γ ⊂ G be an irreducible lattice and π : Γ⟶H be a lattice embedding. Then G ≃ H and there exists an isomorphism π̄ : Ad G⟶Ad H such that the following diagram commutes:

Γ ⟶^{π} π(Γ)
↓^{Ad}  ↓^{Ad}
Ad G ⟶^{π̄} Ad H

Originally, Mostow proved this remarkable theorem for uniform lattices [Mo2]. Mostow’s approach was extended by Prasad [Pr] to encompass non-uniform lattices in rank one groups (and ℚ-rank one lattices). Finally, the remaining cases of nonuniform (irreducible) lattices in higher rank (semi)simple Lie groups were obtained as one of the corollaries of Margulis’ superrigidity [Ma].

The author was partially supported by NSF grant DMS-9803607 and GIF grant G-454-213.06/95.

Motivated by the strong rigidity theorem above, we consider the following general

Problem. Given an (irreducible) lattice Γ in a (semi)simple Lie group G, classify all locally compact groups H which admit lattice embedding, or uniform lattice embedding, of Γ.

Consider the following examples:

Example 1.1. Let Γ be a torsion free subgroup of finite index in SL_{n}(ℤ). The following locally compact groups admit (non-uniform) lattice embeddings of Γ: G = SL_{n}(ℝ); Ad G = PSL_{n}(ℝ); Aut G; groups of the form G′ × K where K is a compact group and G′ is as above; skew-products Aut G ⋉_{Out G} K where Aut G acts through the finite group Out G on K; “almost direct” products: H = (G′ × K)/C, where C is a finite abelian group embedded diagonally in the centers of G′ as above and the center of a compact K. An example of the latter type is H = (SL_{n}(ℝ)×SO_{n})/±I ×I, for even n.

Example 1.2. Let Γ be as above. Obviously it has a uniform lattice embedding (the identity map) in itself; in any discrete Γ′ ⊃ Γ with [Γ′ : Γ] < ∞ (for example in Γ′ = SL_{n}(ℤ) or in PSL_{n}(ℤ)); as well as in direct products Γ′ × K where K is any compact group; or in skew products with compact groups, such as SL_{n}(ℤ) ⋉ 𝕋^{n}, where 𝕋^{n} = ℝ^{n}/ℤ^{n} is the torus. Γ also has uniform lattice embeddings in “almost skew-products”, described by an exact sequence

1⟶F ⟶Γ′ ⋉ K⟶H⟶1

where Γ′ ⊃ Γ (with [Γ′ : Γ] < ∞) acts by automorphisms on a compact group K, and F is a finite abelian group diagonally embedded in the center of Γ′ and as a normal subgroup in K which is pointwise fixed by the Γ′-action.

Roughly speaking, the locally compact groups H in Example 1.1 are built from the ambient Lie group G, while the groups H in Example 1.2 are built from the lattice itself. Non uniform lattices in SL_{2}(ℝ) (i.e. virtually free groups) admit uniform lattice embeddings of a completely different nature:

Example 1.3. Finitely generated non-abelian free groups Γ = F_{k} form (non-uniform) lattices in SL_{2}(ℝ). These groups have uniform lattice embeddings in SL_{2}(ℚ_{p}) and in Aut (T) - the group of automorphisms of the regular 2k-tree T. Taking direct and skew-products with compact groups one obtains additional examples.

The last example suggests the following general

Construction 1.4. Let Γ be a finitely generated group and let X_{Γ,Σ} denote the (unlabeled) Cayley graph of Γ with respect to some finite set Σ of generators. Then Γ has a uniform lattice embedding in the totally disconnected locally compact group Aut (X_{Γ,Σ}) of all automorphisms of this graph. (Indeed, Γ acts simply transitively on the vertices of X_{Γ,Σ}, while the stabilizer of a vertex is a compact subgroup in Aut (X_{Γ,Σ})).

We assert that for lattices in higher rank simple Lie groups the only lattice embeddings are such as the ones described in Examples 1.1 and 1.2:

Theorem A. Let Γ be a lattice in a simple higher rank Lie group G, let H be some locally compact group, which admits a lattice embedding π : Γ⟶H of Γ. Then Λ = π(Γ) is contained in a closed subgroup H_{0} of finite index in H, where H_{0} has one of the following forms:

(1) H_{0} is a central extension of locally compact groups

1⟶C⟶H_{0}⟶^{p}Ad G × K⟶1

where C is a compact abelian group and K is a compact group. More precisely, H_{0} ≅ (G′ ×K′)/C, where the compact abelian group C is diagonally embedded in the centers of a connected locally compact G′ with G′/C ≅ Ad G, and a compact group K′ with K′/C ≅ K. If the universal covering G̃ of G has finite center, one can take C to be a finite abelian group, in which case G′ is a simple Lie group locally isomorphic to G.

(2) H_{0} admits an exact sequence

1⟶F ⟶Γ ⋉ K⟶H_{0}⟶1

where Γ acts by automorphisms on a compact group K, and F is a finite abelian group diagonally embedded in the center of Γ and as a normal subgroup of K, which is elementwise fixed by the Γ-action. If Γ has trivial center, then H_{0} ≅ Γ ⋉ K.

Moreover, denoting by Φ : H_{0}⟶Ad G the (continuous) homomorphism

(1) Φ : H_{0}⟶^{p}Ad G × K⟶^{pr_{1}}Ad G , or

(2) Φ : H_{0}⟶Ad Γ × (K/F)⟶^{pr_{1}}Ad Γ ⊂ Ad G corresponding to the above cases (here pr_{1} is the projection on the first factor) the following diagram commutes

\begin{matrix}Γ & ⟶^{π} & Λ \\ ↓^{Ad} & & ↓^{⊂} \\ Ad G & ⟵^{Φ} & H_{0} \end{matrix}  (1.1)

For uniform lattice embeddings, we have a similar result for lattices in other semisimple Lie groups:

Theorem B. Let Γ be an irreducible lattice in a semisimple Lie group G ≄ SL_{2}(ℝ) and H be some locally compact group which admits a uniform lattice embedding π : Γ⟶H of Γ. Then the conclusions of Theorem A hold.

For uniform lattices in G ≃ SL_{2}(ℝ) similar results hold, except for the commutativity of the diagram (1.1) in case (1). More precisely:

Theorem C. Let Γ be a uniform lattice in G ≃ SL_{2}(ℝ) and let H be a locally compact group, which admits a uniform lattice embedding π : Γ⟶H of Γ. Then Λ = π(Γ) is contained in a closed subgroup H_{0} of finite index in H, where H_{0} has one of the forms (1) or (2) as in Theorem A. Furthermore, if Φ : H_{0}⟶Ad G denotes the (continuous) homomorphism as in Theorem A, then Φ(Λ) ≅ Ad Γ.

In view of the Construction 1.4, one can deduce from Theorems B and C the following

Corollary 1.5. Let Γ be an irreducible lattice in a semisimple Lie group G. In case of G ≃ SL_{2}(ℝ) assume that Γ is uniform in G. Let Σ be some finite generating set for Γ, and let X_{Γ,Σ} denote the corresponding undirected unlabeled Cayley graph. Then X_{Γ,Σ} admits at most finite number of outer automorphisms, i.e.

[Aut (X_{Γ,Σ}) : Γ] < ∞

Moreover, the above index is bounded by some constant i(Γ), which does not depend on the chosen generating set Σ for a given Γ. If Γ ⊂ G ⊂ Aut (G) is not contained in a larger lattice Γ_{∗} ⊂ Aut (G) then i(Γ) = 1.

We refer to the phenomenon described in Theorems A and B as Mostow-Margulis rigidity with locally compact targets. The remaining problem (corresponding to Prasad’s result), of describing locally compact groups H which admit a non-uniform lattice embedding of a rank one lattice Γ, remains open.

Remarks 1.6. (a) Another related problem, posed by Zimmer, is “superrigidity with locally compact targets”, namely classification of locally compact groups H, for which there exists a homomorphism π : Γ⟶H with the image Λ = π(Γ) being sufficiently “dense” in H, where the notion of “density” should replace Zariski density in semisimple Lie groups. Theorem A applies to the situation where π(Γ) forms a lattice in H. Indeed, Margulis’ Normal Subgroup Theorem states that either π(Γ) is finite (so that H is compact), or π factors through an embedding π′ : Γ′⟶H where Γ′ is a lattice with Ad Γ′ ≅ Ad Γ, to which Theorem A applies.

(b) In principle, using structure Theorems A (respectively B), one can also classify not only the targets H of (uniform) lattice embeddings of Γ, but rather (uniform) lattice embeddings π : Γ⟶H themselves, up to Aut H. We have not addressed this question here.

(c) The proofs of Theorems A and B give effective bounds on the index [H : H_{0}] in terms of |Out (G)| < ∞ in case (1), and the index of Ad Γ in a maximal lattice Γ_{∗} in Aut (Ad G), containing Ad Γ in case (2). The latter gives the bound i(Γ) on the index [Aut (X_{Γ,Σ}) : Γ] in Corollary 1.5.

(d) In Section 3.6 we show that the description of H_{0} in case (1) of Theorem C, as an almost direct product H_{0} ≅ (G′ × K)/C over a compact abelian center C, cannot be replaced by an almost direct product over a finite abelian center C (as it is claimed in Theorems A and B for the case where the universal covering G̃ has a finite center). However, it is unclear to the author, whether compact center C can be replaced by a finite center in case (1) of Theorems A, B.

(e) Theorem C leaves open the problem of the description of locally compact groups H which contain a uniform lattice Γ, where Γ ⊂ G ≃ SL_{2}(ℝ) is a non-uniform lattice. Such groups Γ contain a finitely generated non-abelian free group F_{k} as a finite index subgroup. So the question reduces to a description of locally compact groups H which contain F_{k} is a cocompact lattice. As one of the corollaries of their study of quasi-actions on trees Lee Mosher, Michah Sageev and Kevin Whyte [MSW] recently proved that such a group H admits a short exact sequence {1} → K → H → L → {1}, where K is a compact group and L ⊆ Aut (T) is a closed cocompact subgroup of isometries of a tree T which is quasi-isometric to a regular tree (and to F_{k}).

About the proofs. Having very similar appearance, Theorems A and B, C have quite different proofs. The proof of Theorem A uses measure-theoretic aspects of (semi)simple Lie group actions, such as Zimmer’s superrigidity for cocycles (subsection 2.1) which is a generalization of Margulis’ superrigidity, and an argument (subsection 2.2), which has a lot in common with the smoothness of algebraic actions on spaces of measures. On the other hand, the proof of Theorem B (and that of C) is based on, by now well developed, theory of quasi-isometries, which has Mostow’s proof of strong rigidity as one of its origins. The proof of Theorem C also uses some special features of the group of homeomorphisms of the circle.

Acknowledgments. Many thanks to Benson Farb, Alex Eskin and Rich Schwartz for many inspiring discussions on geometry and rigidity of lattices, to Etienne Ghys for introducing me to Homeo (S^{1}), and to Marc Bourdon for the elegant argument 3.8. I would also like to thank Anatole Katok for his support and encouragement during the year that I spent at the Pennsylvania State University, being a Post Doctoral Fellow at the Center for Dynamical Systems. Part of this work was done when I enjoyed the hospitality of the University of Bielefeld, Germany.

2. Proof of Theorem A

Outline of the proof. The essential idea of the proof is to establish a homomorphism Φ : H⟶Aut (Ad G), for which the following diagram commutes:

Γ  ⟶^{π} Λ
↓^{Ad}  ↓^{⊂}
Aut (Ad G) ⟵^{Φ} H

(2.1)

This is done in two steps:

• Using superrigidity for measurable cocycles, a measurable bi-Γ-equivariant map Φ : H⟶Aut (Ad G), satisfying (2.1), is constructed (subsection 2.1).

• It is proved that any such bi-Γ-equivariant measurable map coincides a.e. with

a continuous homomorphism (subsection 2.2). The image Φ(H) ⊆ Aut (Ad G) is shown either to contain Ad G or to form a lattice Γ_{∗} ⊇ Ad Γ in Aut (Ad G), in which case [Γ_{∗} : Γ] < ∞ (subsection 2.3), while the kernel K = Ker(Φ) is a compact normal subgroup in H. Passing to the finite index subgroup H_{0} := Φ^{−1}(Ad G) or H_{0} := Φ^{−1}(Ad Γ), descriptions (1) or (2) are obtained in subsection 2.4.

2.1. Construction of a measurable equivariant map Φ : H → Aut (Ad G).

Theorem 2.1. There exists a unique measurable map Φ : H → Aut (Ad G) such that

Φ(π(γ_{1}) h π(γ_{2})) = (Ad γ_{1}) (Φ(h)) (Ad γ_{2})  (2.2)

for every γ_{1}, γ_{2} ∈ Γ and a.e. h ∈ H.

The group structure of H is immaterial for this statement. It holds whenever H is an (infinite) measure space with two commuting, measure preserving, free, finite covolume actions of a higher rank lattice Γ (here the actions are from the left and from the right). This was proved in [F1] Theorem 4.1. For the sake of completeness we give a

Sketch of the proof. Consider the finite measure Lebesgue space H/Λ with the measure preserving (transitive) left H-action on it. Choose a (Lebesgue) measurable cross section s : H/Λ⟶H for the natural projection H⟶H/Λ, and define a measurable cocycle

α_{0} : H × (H/Λ)⟶Λ, by α_{0}(h_{1}, hΛ) := s(h_{1}hΛ)^{−1}h_{1} s(hΛ)

Let α : Γ × (H/Λ)⟶Ad Γ ⊂ Ad G be the measurable cocycle, defined by

α(γ, hΛ) := Ad ∘ π^{−1} (α_{0}(π(γ), hΛ))

Working separately on each of the Γ-ergodic components, one checks that α is Zariski dense in Ad G ([F1] Lemma 4.2, see also [Zi] p. 99). At this point we use the assumption that G is a simple Lie group of higher rank to apply the superrigidity for cocycles theorem ([Zi] 5.2.5), which gives an existence of a measurable ϕ : H/Λ⟶Ad G and a homomorphism ρ : Γ → Ad G such that

α(γ, hΛ) = ϕ(π(γ)hΛ)^{−1}ρ(γ)ϕ(hΛ)

In the above formula, ρ(γ) and ϕ(hΛ) can be replaced by g^{−1}ρ(γ) g and g^{−1} ϕ(hΛ) for any g ∈ Ad G. Allowing ϕ to take values in Aut (Ad G) ⊃ Ad G (which enables to use g ∈ Aut g), and using Margulis’ superrigidity ([Ma]), one can always assume that ρ(γ) = Ad γ. Fixing ρ this way, the map ϕ turns out to be uniquely determined (Remark 2.4.(a)). Now one reassembles the definition of ϕ on the ergodic components to a single, still measurable, map ϕ : H/Λ⟶Aut (Ad G). Extending ϕ to Φ : H⟶Aut (Ad G) in a Γ-equivariant (from the right) way, namely by

Φ(h) := ϕ(hΛ) Ad (s(hΛ)^{−1}h)

one verifies that Φ satisfies the relation (2.2).

Remark 2.2. The above argument is the only place in the proof, where the assumption that G is a higher rank simple Lie group, is used. To apply this construction to an irreducible lattice Γ in a higher rank semisimple G = ΠG_{i}, one needs to know that each of the simple factors G_{i} acts ergodically on the skew-product G/Γ×X, where X is a Λ-ergodic component of H/Λ. This would follow, for example, if one knew that Λ is mixing on each of the Λ-ergodic components of H/Λ.

2.2. Equivariant measurable map Φ is a continuous homomorphism. The following Theorem is formulated for our particular setup, but it might have an independent interest:

Theorem 2.3. Let G be a semisimple Lie group, H a locally compact group, and Λ ⊆ H a lattice in H. Assume that there exists a homomorphism ρ : Λ⟶Ad G with Zariski dense image, and a measurable map Φ : H⟶Aut (Ad G) which satisfies

Φ(λ_{1} h λ_{2}) = ρ(λ_{1}) Φ(h) ρ(λ_{2})

for all λ_{1}, λ_{2} ∈ Λ and m_{H}-a.e. h ∈ H. Then the map Φ coincides m_{H}-a.e. with a continuous homomorphism Φ_{0} : H⟶Aut (Ad G) with Φ_{0}|_{Λ} = ρ, the image L = Φ_{0}(H) is a closed subgroup of Aut (Ad G) for which the pushforward measure Φ_{∗}(m_{H}) gives the Haar measure m_{L} on L.

Proof. Consider the measurable function F : H × H⟶Aut (Ad G) defined by

F (h_{1}, h_{2}) := Φ(h_{1}) Φ(h_{1}^{−1}h_{2}) Φ(h_{2})^{−1}

Observe that for any λ ∈ Λ and a.e. h_{1}, h_{2} one has

\begin{aligned}F (h_{1}λ, h_{2}) & = & F (h_{1}, h_{2}) \\ F (h_{1}, h_{2}λ) & = & F (h_{1}, h_{2}) \\ F (λ h_{1}, λ h_{2}) & = & ρ(λ) F (h_{1}, h_{2}) ρ(λ)^{−1} \end{aligned}  (2.3)

Hence F descends to a measurable function f on a probability space X := H/Λ×H/Λ. This f satisfies

f(λ · x) = ρ(λ) f(x)ρ(λ)^{−1}

where λ : x ↦ λ · x denotes the measure preserving diagonal left Λ-action on H/Λ × H/Λ.

Let μ := f_{∗}(m_{H/Λ} × m_{H/Λ}) denote the push forward measure on Aut (Ad G). Then μ is a probability measure which is invariant under conjugation by elements of the Zariski dense subgroup ρ(Λ). We claim that such μ has to be concentrated on the identity: μ = δ_{e}. Indeed, it is well known that if g is a regular semisimple element of Ad G ⊂ Aut (Ad G), and g′ ∈ Aut (Ad G) does not commute with g, then g^{n}g′g^{−n} → ∞ for n → ∞ or n → −∞. Poincaré recurrence (for μ under conjugation by g) implies that μ is supported on the centralizer of g. Since regular elements of ρ(Λ) are Zariski dense in Ad G, we conclude that μ is supported on the center of Aut (Ad G), which is trivial.

Hence almost everywhere F (h_{1}, h_{2}) = e, so that

Φ(h_{1}^{−1}h_{2}) = Φ(h_{1})^{−1}Φ(h_{2}), a.e. on H × H

This means that Φ is an a.e. homomorphism and Φ(H) is an a.e. group with respect to the pushforward measure m_{∗} = Φ_{∗}(m_{H}). This is known to imply (cf. [Zi] Appendix B) that m_{∗} is the Haar measure m_{L} of a closed subgroup L ⊆ Aut (Ad G) and Φ(h) = Φ_{0}(h) a.e. where Φ_{0} : H⟶L ⊆ Aut (Ad G) is a continuous homomorphism. In particular, for a fixed λ ∈ Λ and a.e. h ∈ H:

ρ(λ) Φ(h) = Φ(λh) = Φ_{0}(λh) = Φ_{0}(λ) Φ_{0}(h) = Φ_{0}(λ) Φ(h)

so that Φ_{0}(λ) = ρ(λ) for λ ∈ Λ.

Remarks 2.4. (a) Under the conditions of Theorem 2.3 with a fixed ρ : Λ⟶Γ there exists at most one, up to m_{H}-null sets, bi-equivariant measurable map Φ, and in particular a unique extension Φ_{0} of ρ. The proof is similar to the above one: if Φ_{1}, Φ_{2} are two bi-equivariant measurable maps, then F (h) := Φ_{1}(h) Φ_{2}(h)^{−1} has properties similar to (2.3), so that F_{∗}(m_{H/Λ}) is a probability measure on Aut (Ad G) which is invariant under the conjugation by g ∈ ρ(Λ), and hence is δ_{e}.

(b) Consider the following example: let G = H = PSL_{2}(ℝ), Λ_{1} be an infinite cyclic subgroup of the diagonal group, Λ_{2} ⊂ G be a uniform lattice (i.e. a surface group) and ρ = Id. The Λ_{1} ≅ ℤ-action on G/Λ_{2} is a (discrete time) map of the geodesic flow on the unit tangent bundle SM to the Riemann surface M = K\G/Λ_{2} equipped with the standard measure. Such an action is measurably isomorphic to a Bernoulli shift, which is known to have a vast family of measurable automorphisms. These automorphisms give rise to many measurable maps Φ : G⟶G which satisfy a.e. on G:

Φ(λ_{1}gλ_{2}) = λ_{1} Φ(g) λ_{2}, (λ_{1} ∈ Λ_{1}, λ_{2} ∈ Λ_{2})

This example shows that the conclusion of Theorem 2.3 does not follow just from the ergodicity of the ρ(Λ_{1}) × ρ(Λ_{2})-action on G.

Applying Theorem 2.3 with ρ = π^{−1} to the measurable map Φ : H⟶Aut (Ad G), constructed in Theorem 2.1, we conclude that Φ (possibly, after an adjustment on a null set) is a continuous homomorphism satisfying

Φ ∘ π = Ad : Γ ⟶ Ad Γ ⊂ Ad G ⊂ Aut (Ad G)  (2.4)

Hereafter Φ denotes this continuous homomorphism.

2.3. The Image and the Kernel of Φ : H → Aut (Ad G). Since Ad Γ is Zariski dense in Ad G the connected component of the closed subgroup Φ(H) ⊆ Aut (Ad G), which contains Ad Γ is either Ad G, or trivial. In the former case Ad G has finite index in Φ(H) (recall that [Aut (Ad G) : Ad G] < ∞), and in the latter case Φ(H) is a discrete group containing Ad Γ, necessarily as a finite index subgroup. We shall now restrict our attention to the closed subgroup H_{0} ⊆ H of finite index, defined by H_{0} := Φ^{−1}(Ad G) or H_{0} := Φ^{−1}(Ad Γ), corresponding to the above cases. Note also that (2.4) shows that Λ ⊆ H_{0}.

Next we claim that the kernel K = Ker(Φ) ◁ H_{0} is compact. First consider the case of Φ(H_{0}) = Ad G. Let F ⊂ Ad G be a measurable subset which forms an Ad Γ-fundamental domain. Observe that the equivariance of Φ implies that E := Φ^{−1}(F) forms a Λ-fundamental domain in H_{0}, and we can normalize the Haar measure m_{H_{0}} (and hence m_{∗}) so that m_{H_{0}}(E) = m_{∗}(F) = 1. Disintegration of m_{H_{0}}|_{E} (and then all of m_{H_{0}}) with respect to m_{∗}|_{F} (respectively m_{∗}) gives a family of probability measures {μ_{g}}_{g∈Ad G} supported on the fibers K_{g} := Φ^{−1}({g}) which are K-cosets. The uniqueness of the Haar measures (normalized as above) as (left) invariant measures, together with the uniqueness of the disintegration procedure, give for every h ∈ H and m_{∗}-a.e. g ∈ Ad G:

h μ_{g} = μ_{Φ(h)g}

In particular for each k ∈ K and m_{∗}-a.e. g, one has k μ_{g} = μ_{g}, and, by a standard Fubini argument, one concludes that μ_{e} is a left K-invariant probability measure on K, i.e. K is compact.

The case of Φ(H_{0}) = Ad Γ is even simpler: the measure on the fiver K = Φ^{−1}({e_{Ad Γ}}) is finite and K-invariant, so K is compact.

2.4. The structure of H_{0}. By now we have shown that, given π : Γ⟶Λ ⊆ H, there exists a (unique) continuous homomorphism Φ : H⟶Aut (Ad G) satisfying (2.1). Moreover, H_{0} is described either by the exact sequence

1⟶K⟶H_{0}⟶^{Φ}Ad G⟶1  (2.5)

or by the exact sequence

1⟶K⟶H_{0}⟶^{Φ}Ad Γ⟶1  (2.6)

In both cases K denotes a compact group.

Case (1): Φ(H_{0}) = Ad G.

Theorem 2.5. Suppose that a locally compact H admits an exact sequence of continuous homomorphisms

1⟶K⟶H⟶^{Φ}L⟶1

where K is a compact normal subgroup of H and L is a connected group. Then H contains a closed normal connected subgroup L′, which centralizes K, and forms a central extension of L by a compact abelian group A := L′ ∩ K:

1⟶A⟶L′⟶^{Φ}L⟶1

The group H is then described by the exact sequence

1⟶A⟶^{j}L′ × K⟶H⟶1

where j(a) := (a, −a) is a diagonal embedding of the compact abelian A in the centers of L′ and K.

Proof. Let us denote by C_{H}(K) the centralizer of K in H, i.e.

C_{H}(K) := {h ∈ H | [h, k] = 1 ∀k ∈ K}

C_{H}(K) is a closed normal subgroup of H. We claim that H = K · C_{H}(K).

Let us assume first that K has a faithful finite dimensional representation, i.e. it is a subgroup of some U(n). In this case Out K is known to be discrete. The conjugations h : k ↦ h^{−1}k h gives a continuous homomorphism

i : H⟶Aut K which descends to j : L⟶Out K

Since L is connected, this j must be trivial, so i(H) = i(K) = Inn K ⊆ Aut K. Note that i(h) = 1 iff h lies in the centralizer C_{H}(K) of K in H. Hence H = K · C_{H}(K) as asserted.

Now consider the general case of a compact K. Let π ∈ K̂ be an irreducible (unitary) K-representation. The H-action on K moves π to π^{h} ∈ K̂ defined by

π^{h}(k) := π(h^{−1}k h) (k ∈ K, h ∈ H, π ∈ K̂)

The corresponding continuous H-action on the characters χ_{π}(k) = Tr π(k) is trivial on K, and therefore descends to an action of L. Since {χ_{π}}_{π∈K̂} are orthogonal, the latter action of the connected group L is trivial. Thus χ_{π^{h}} = χ_{π}, implying π^{h} ∼ π. Hence Ker(π) is normal not only in K but also in H.

Let us enumerate by π_{i}, i = 0, 1, … all the irreducible K-representations and form the increasing sequence of finite-dimensional unitary K-representations

σ_{n} = ⊕_{i=0}^{n}π_{i}

Since Ker(σ_{n}) = ⋂_{0}^{n} Ker(π_{i}) is a closed normal subgroup of H, the exact sequence of (continuous homomorphisms)

1⟶K⟶H⟶L⟶1

descends to the exact sequence

1⟶K_{n}⟶H_{n}⟶L⟶1

where K_{n} = K/Ker(σ_{n}) and H_{n} = H/Ker(σ_{n}). The discussion of the finite dimensional case above yields that H_{n} = K_{n} · C_{H_{n}}(K_{n}). This means that given a fixed element h ∈ H we can find k_{n} ∈ K and z_{n} ∈ H so that

h = k_{n} z_{n} and ∀k ∈ K : [z_{n}, k] ∈ Ker(σ_{n})

Passing to a subsequence n_{i}, we can assume that k_{n_{i}} → k_{∗} ∈ K and hence z_{n_{i}} → z_{∗} = h k_{∗}^{−1} ∈ H. Since Ker(σ_{n_{i}}) form a decreasing sequence of compact subsets, we have

h = k_{∗} z_{∗} and ∀k ∈ K : [z_{∗}, k] ∈ Ker(σ_{n_{i}}), ∀i

Recall that {π_{n}}_{n} separate points of K, and so do {σ_{n_{i}}}_{i}, hence ⋂_{i} Ker(σ_{n_{i}}) = {1} and therefore z_{∗} ∈ C_{H}(K). This shows that H = K · C_{H}(K).

Now let C denote the center of K, i.e. C = K ∩ C_{H}(K). Observe that C_{H}(K) is a central extension of L by C:

L = H/K = K · C_{H}(K)/K ≅ C_{H}(K)/K ∩ C_{H}(K) = C_{H}(K)/C

Taking L′ to be the connected component of C_{H}(K) and A := L′ ∩ K one obtains the assertion of the Theorem.

Applying Theorem 2.5 to the sequence (2.5) one obtains the first assertion of Theorem A, case (1). In the case of G with a finite universal covering G̃, the description of H_{0} as (G′ × K′)/C where a compact C is diagonally embedded in the centers of G′ and K′, can be reduced to a finite C, using the following:

Lemma 2.6. Let G′ be a locally compact group which is a (topological) central extension

1⟶C⟶L⟶^{p}G⟶1  (2.7)

of a semisimple Lie group G by a compact abelian group C. Assume that the center of G is trivial, and the center Z of its universal covering G̃ is finite. Then there exists a closed connected subgroup G′ ⊆ L with a finite center Z′ = G′ ∩ C, so that the finite covering

1⟶Z′⟶G′⟶^{p}G⟶1

is a local isomorphism of the semisimple Lie groups G′ ≃ G.

Proof. First consider the situation, where C is totally disconnected. Then (2.7) is a fiber bundle with a totally disconnected fiber C, over the base G which is a connected manifold. Hence any point of G can be connected by a path in G, starting at e_{G}; such path has a unique lifting to a path in L, starting at e_{L}; while homotopic paths in G lift to homotopic paths in L. Define G′ to be the path-connected component of the identity e_{L} in L. Then G′ ∩ C is a homomorphic image of π_{1}(G, e_{G}) = Z which is finite. Hence G′ is a finite covering of G. It is closed in L, since the covering map p|_{G′} : G′⟶G is finite to one.

Next consider the general case of a compact C. The connected component C_{0} of the identity in C, is a closed (compact) subgroup of the center of L. Dividing (2.7) by C_{0}, one arrives in the situation discussed above, in which case L/C_{0} contains a closed connected finite covering G′_{0} of G. Let L′ ⊆ L be the preimage of G′_{0} ⊆ L/C_{0}. It is a closed connected subgroup in L, which is a topological central extension

1⟶C_{0}⟶L′⟶^{p′}G′_{0}⟶1

of the semisimple Lie group G′_{0} by a connected compact group C_{0}. This central extension is an inverse limit of central extensions of Lie groups

1⟶C_{0,n}⟶L′_{n}⟶^{p′_{n}}G′_{0}⟶1

where C_{0,n} are finite dimensional tori. Locally, these extensions are trivial, since the second cohomology H^{2}(Lie(G′_{0}), ℝ^{d}) vanishes. Hence L′_{n} contains a closed subgroup G′_{n} for which p′_{n} : G′_{n}⟶G′_{0} is a finite covering. Since the degree of this covering is bounded by |Z| < ∞, the subgroups G′_{n} stabilize (in the sense that G′_{n+1}⟶G′_{n} is one-to-one for large n), so that the limit group L′ contains a closed subgroup G′ for which p′ : G′⟶G′_{0} is a finite covering.

This completes the proof of Theorem A for Case (1).

Case (2): Φ(H_{0}) = Ad Γ. The compact group K is normal in H_{0} ⊃ Λ ≅ Γ, so Γ acts on K by automorphisms:

γ : k ↦ k^{γ}, where k^{γ} := π(γ)^{−1} k π(γ)

The corresponding skew product Γ ⋉ K maps homomorphically into H_{0} by

p : Γ ⋉ K⟶H_{0} defined by p(γ, k) := π(γ) · k

Since the homomorphism

Γ ⋉ K⟶^{p}H_{0}⟶^{Φ}Ad Γ

coincides with the natural homomorphism

Γ ⋉ K⟶Γ⟶^{Ad}Ad Γ

it follows that p is onto, and that Z = Ker(p) consists of elements z = (γ_{z}, k_{z}) with γ_{z} ∈ Ker(Ad : Γ⟶Ad Γ) - the finite abelian center of Γ, while π(γ_{z}) = k_{z}^{−1} ∈ Λ ∩ K. Observe that

• K_{Z} := {k_{z} ∈ K | z ∈ Z} is a finite abelian group, isomorphic to Z;

• K_{Z} is normal in K, because {1} × K and Z are normal in Γ ⋉ K, so that

{1} × K_{Z} = ({1} × K) ∩ Z ◁ Γ ⋉ K

• for each γ ∈ Γ, one has γ^{−1}γ_{z}γ = γ_{z}, so that k_{z}^{γ} = k_{z}, for each z ∈ Z. Hence H_{0} is described by the exact sequence

1⟶Z⟶^{j}Γ ⋉ K⟶^{p}H_{0}⟶1

with j(z) = (γ_{z}, k_{z}), where z ↦ γ_{z} is an embedding of Z in the finite abelian center of Γ; and K_{Z} = {k_{z}}_{z∈Z} forms a normal subgroup in K whose elements are fixed by the action of Γ.

Clearly, if Γ has trivial center, then H_{0} is just isomorphic to the skew-product H_{0} ≅ Γ ⋉ K.

The proof of Theorem A is completed.

3. Proof of Theorems B and C

Outline of the proof. As in the proof of Theorem A, the essential step is to construct a continuous homomorphism Φ : H⟶Aut (Ad G). If G ≄ SL_{2}(ℝ) the constructed Φ will satisfy (2.1); while in the case G ≃ SL_{2}(ℝ) we shall only obtain that

Φ ∘ π : Γ⟶Aut (Ad G)

takes Γ to a lattice Γ′ ⊂ PSL_{2}(ℝ) ⊂ Aut (Ad G), isomorphic to Ad Γ.

The homomorphism Φ is constructed using the quasi-isometry groups. For any finitely generated group Γ, there is an associated quasi-isometry group QI (Γ), which we shall consider as an abstract group with no topology. The quasi-isometry group has the property that any uniform lattice embedding π of Γ in a locally compact group H, gives rise to a homomorphism (of abstract groups) Φ_{π} : H⟶QI (Γ) (see subsection 3.2).

Consider the case where Γ is a uniform lattice in a simple Lie group G of rank one. Then Γ is quasi-isometric to the symmetric space X = G/K, which has strictly negative Riemannian curvature (X is real, complex, quaternionic hyperbolic space or the Cayley plane), in which case there exists an embedding

β : QI (Γ)⟶Homeo (∂X)

in the group of homeomorphisms of the boundary ∂X of X. This gives an abstract homomorphism of groups Ψ_{π} : H⟶Homeo (∂X)

Ψ_{π} : H⟶^{Φ_{π}}QI (Γ)⟶^{β}Homeo (∂X)

which turns out to be continuous (Theorem 3.5). The given uniform lattice embedding σ : Γ⟶G gives rise to the homomorphism of abstract groups

Φ_{σ} : Aut (Ad G)⟶QI (Ad Γ) ≅ QI (Γ)

and to another continuous homomorphism

Ψ_{σ} : Aut (Ad G)⟶^{Φ_{σ}}QI (Ad Γ)⟶^{β}Homeo (∂X)

which is known to form a continuous isomorphic embedding with a continuous inverse Ψ_{σ}^{−1}. The following is crucial Claim 3.1. The images of Ψ_{π}(H) and Ψ_{σ}(Aut (Ad G)) in Homeo (∂X), are locally compact groups, which contain Ψ_{π}(Λ) and Ψ_{σ}(Ad Γ) as uniform lattices. From the construction one has

Ψ_{π}(π(γ)) = Ψ_{σ}(Ad γ), (γ ∈ Γ)

Moreover, if G ≄ SL_{2}(ℝ) then Ψ_{π}(H) ⊆ Ψ_{σ}(Aut (Ad G)) and

Φ := Ψ_{σ}^{−1} ∘ Ψ_{π} : H⟶Homeo (∂X)⟶Aut (Ad G)

is a continuous homomorphism, satisfying (1.1).

In the case of G ≃ SL_{2}(ℝ), there exists an f ∈ Homeo (∂X) such that

f^{−1} Ψ_{π}(H) f ⊆ Ψ_{σ}(Aut (Ad G))

and the continuous homomorphisms

Φ := Ψ_{σ} ∘ f^{−1}Ψ_{π} f : H⟶Aut (Ad G)

maps Λ = π(Γ) onto a lattice Φ(Λ) ⊂ Aut (Ad G), which is isomorphic (but not necessarily conjugate) to Ad Γ.

From this claim the proof of Theorems B and C can be completed as in subsections 2.3 and 2.4.

3.1. Quasi-isometries. Recall the notion of quasi-isometries (see [Gr], [GP] for detailed discussions): a map q : X_{1}⟶X_{2} between proper metric spaces (X_{1}, d_{1}) and (X_{2}, d_{2}) is said to be a quasi-isometric embedding if there exist constants M, A such that for all x, y ∈ X_{1}:

\frac{1}{M} · d_{1}(x, y) − A ≤ d_{2}(q(x), q(y)) ≤ M · d_{1}(x, y) + A  (3.1)

Such q can be called an (M, A)-quasi-isometric embedding to emphasize that (3.1) is satisfied for specific M and A.

A quasi-isometric embedding q : X_{1}⟶X_{2} with an image q(X_{1}) which is within bounded distance from all of X_{2}, is called a quasi-isometry, and the spaces (X_{1}, d_{1}), (X_{2}, d_{2}) are said to be quasi-isometric. Two quasi-isometries (or quasi-isometric embeddings) q, q′ : X_{1}⟶X_{2} are said to be equivalent (notation: q ∼ q′) if the distance

D(q, q′) := sup{d_{2}(q(x), q′(x)) | x ∈ X_{1}}

is finite. Now consider the collection of all self quasi-isometries of a fixed proper metric space (X, d). It forms a semi-group with respect to composition (which respects the equivalence relation ∼). Modulo this relation, the semi-group of actual quasiisometries becomes a group, called the quasi-isometry group of (X, d) and is denoted by QI (X, d). Quasi-isometric spaces have isomorphic quasi-isometry groups. Therefore, for a finitely generated group Γ it makes sense to talk about its quasi-isometry group QI (Γ) without specifying any particular left invariant word metric. Note also, that the isometric left Γ-action on itself (with respect to some/any left-invariant word metric) defines a natural homomorphism

ρ : Γ⟶QI (Γ)

3.2. Uniform lattice embeddings and quasi-isometries. We claim that the notions of quasi-isometries appear naturally in our situation, where a given finitely generated group Γ has an embedding π : Γ → H as a uniform lattice in an (unknown) locally compact group H. There are two different natural constructions both giving rise to a homomorphism (of abstract groups)

H⟶QI (Γ)

the restriction of which to Γ coincides with the standard homomorphism ρ : Γ → QI (Γ). The first of the two is the following:

Construction 3.2. Choose some open subset E ⊂ H with compact closure, such that H = ∪_{γ∈Γ}π(γ)E, and fix some (typically discontinuous) function p : H⟶Γ, satisfying h ∈ π(p(h)) E for h ∈ H. With this choice of E and p define a family of maps {q_{h} : Γ⟶Γ}_{h∈H} by the rule q_{h}(γ) := p(hπ(γ)). If E is a neighborhood of the unit e, we can choose p with p(π(γ)) = γ, so that

q_{γ}(γ′) = γγ′  (3.2)

Lemma 3.3. Let E, p and {q_{h}}_{h∈H} be as above. Then

(a) Each q_{h} : Γ⟶Γ is a quasi-isometry of Γ and its equivalence class [q_{h}] ∈ QI (Γ) depends only on h ∈ H (and not on the choice of E and p).

(b) The map Φ_{π} : H⟶QI (Γ), given by Φ_{π}(h) = [q_{h}], is a homomorphism of (abstract) groups, such that the composition

Γ⟶^{π}H⟶^{Φ_{π}}QI (Γ)

coincides with the standard homomorphism ρ : Γ⟶QI (Γ).

(c) {q_{h}}_{h∈H} are (M, A)-quasi-isometries for some fixed M and A, depending just on E, p, and independent of h ∈ H.

(d) There exists a constant B with the following property: given any finite set F ⊂ Γ there is a neighborhood of the identity V ⊂ H such that d(q_{h}(γ), γ) ≤ B for all γ ∈ F and h ∈ V.

Remarks 3.4. (a) The above construction is a particular case of Gromov’s topological equivalence, where one considers two finitely generated groups Γ and Γ′ with commuting, proper, cocompact actions on a locally compact space X (in our special case Γ = Γ′, X = H and the commuting actions are from the left and from the right). In this general setup, each point x ∈ X defines a quasi-isometry q_{x} : Γ⟶Γ′ so that properties (a) and (c) of the Lemma still hold. If H forms a locally compact group, one obtains in addition the homomorphism (b) satisfying (d).

(b) In general the collection of actual (M, A)-quasi-isometries {q_{h}}_{h∈H} is not closed under composition: q_{h_{1}} ∘ q_{h_{2}} is equivalent, but typically is not equal, to q_{h_{1}h_{2}}.

An alternative construction of a homomorphism H⟶QI (Γ) uses just the left Γaction on H and a choice of a left invariant metric on H, as follows: fix a proper left invariant metric d on H (here “proper” means that the closed balls with respect to d are compact subsets of H). Actually it suffices to use a proper pseudo-metric such as d(h_{1}, h_{2}) = min{n | h_{2}^{−1}h_{1} ∈ B^{n}} where B ⊆ H is a symmetric compact neighborhood of the identity which generates H (such sets exists because H is compactly generated, since it contains a finitely generated cocompact lattice). One can show that the restriction of such d to π(Γ) is quasi-isometric to a word metric d_{w} on Γ, and (H, d) becomes quasi-isometric to (Γ, d_{w}). This gives rise to a homomorphism

H⟶QI (H) ≅ QI (Γ)

which has the properties (b), (c), (d) of the first construction.

However we shall work with the construction 3.2 which is conceptually closer to the ideas in Section 2.1.

Proof of Lemma 3.3 Fix some finite set S of generators for Γ, and let d denote the corresponding left invariant word metric on Γ. Observe that, for any two compact sets Q, Q′ ⊂ H there is at most finite number of γ ∈ Γ, so that Q π(γ) ∩ Q′ ≠ ∅. This implies that for any sets E, E′ ⊂ H with compact closures, there exists an integer A = A(E, E′), so that

π(γ) E ∩ π(γ′) E′ ≠ ∅ ⟹ d(γ, γ′) ≤ A  (3.3)

In particular, if q_{h} and q′_{h} are constructed from E, p and E′, p′ respectively, then for some A = A(E, E′) any fixed h ∈ H and all γ ∈ Γ:

h π(γ) ∈ π(q_{h}(γ)) E ∩ π(q′_{h}(γ))E′ ⟹ d(q_{h}(γ), q′_{h}(γ)) ≤ A  (3.4)

Denote by B_{n} the n-ball in Γ centered at the origin. The set E B_{1} = ∪_{γ∈B_{1}}E π(γ) has a compact closure in H, and therefore E B_{1} ⊂ B_{M} E for some sufficiently large integer M = M(E). Then for all n > 1, we have:

E π(B_{n}) ⊂ π(B_{M}) E π(B_{n−1}) ⊂ ⋯ ⊂ π(B_{M·n})E

Take some fixed h ∈ H and any γ_{1}, γ_{2} ∈ Γ. Denoting n = d(γ_{1}, γ_{2}), we have

h π(γ_{1}) ∈ q_{h}(γ_{1}) E  and h π(γ_{2}) ∈ q_{h}(γ_{2}) E
h π(γ_{2}) = h π(γ_{1} (γ_{1}^{−1}γ_{2})) ∈ π(q_{h}(γ_{1})) E π(γ_{1}^{−1}γ_{2})
⊂ π(q_{h}(γ_{1})) E π(B_{n}) ⊂ π(q_{h}(γ_{1})B_{M·n})E

Thus π(q_{h}(γ_{2})) E intersects π(q_{h}(γ_{1})B_{M·n})E, and using (3.3), we deduce that

d(q_{h}(γ_{1}), q_{h}(γ_{2})) ≤ M · d(γ_{1}, γ_{2}) + A

Similar arguments show the other properties of quasi-isometries. Hence q_{h} is a quasiisometry of Γ. Its class [q_{h}] ∈ QI (Γ), which we shall denote by Φ_{π}(h), does not depend on the choice of E, p as (3.4) shows. This proves (a).

Next take h_{1}, h_{2} ∈ H and observe that for each γ ∈ Γ, h_{i} π(γ) ∈ π(q_{h_{i}}(γ)) E for i = 1, 2, and therefore

(h_{1} h_{2}) π(γ) ∈ h_{1} π(q_{h_{2}}(γ)) E ⊂ π(q_{h_{1}}(q_{h_{2}}(γ))) E · E

Since E′ := E · E has a compact closure in H, (3.3) implies that

q_{h_{1}} ∘ q_{h_{2}} ∼ q_{h_{1}h_{2}} i.e. Φ_{π}(h_{1}) · Φ_{π}(h_{2}) = Φ(h_{1}h_{2})

Similarly q_{h} ∘ q_{h^{−1}} ∼ Id_{Γ}, so that Φ_{π} is a homomorphism of abstract groups. Note also, that since Φ_{π} does not depend on the choice of E, p, we could choose them so that (3.2) holds. This shows that Φ_{π} ∘ π = ρ. Hence (b) is proved.

Statement (c) follows from the proof of (a) above, where M and A were constructed independently of h ∈ H.

Next fix some neighborhood U of the identity e_{H} which has a compact closure. Then E ∪ U has a compact closure Q, and there is a constant B = A(Q) such that

γQ ∩ γ′Q ≠ ∅ ⟹ d(γ, γ′) ≤ B

Given any finite set F ⊂ Γ there is an open neighborhood V of e_{H} such that π(γ)^{−1}V π(γ) ⊂ U for all γ ∈ F, and therefore V π(γ) ⊆ π(γ)U for γ ∈ F. For each h ∈ V one also has hπ(γ) ∈ π(q_{h}(γ)) E, which yields π(q_{h}(γ))Q ∩ π(γ)Q is not empty for γ ∈ F, h ∈ V, so that d(q_{h}(γ), γ) ≤ B for all γ ∈ F, h ∈ V. Since B is independent of F, property (d) is proved.

In our particular situation Γ is a lattice in a (semi)simple Lie group G. Quasiisometries of lattices have been thoroughly analyzed by several people over a series of many papers, and by now a comprehensive understanding of these objects has been achieved. The reader is referred to the survey papers by Farb [Fa] and by Gromov and Pansu [GP] for details and further references for this beautiful theory. In what follows we shall only briefly mention some of the facts needed for the proof of Theorems B and C. We need to consider several cases.

3.3. Γ is a uniform lattice in a simple G of rank one. A uniform lattice Γ in a rank one simple Lie group G forms a hyperbolic group in the sense of Gromov. Given a general Gromov hyperbolic group Γ there is an associated compact metric space ∂Γ - the boundary of Γ - equipped with a continuous (essentially faithful) Γ-action by homeomorphisms. The group QI (Γ) has a homomorphism (in fact an embedding)

β : QI (Γ)⟶Homeo (∂Γ)

such that the composition Γ⟶^{ρ}QI (Γ)⟶^{β}Homeo (∂Γ) corresponds to the standard Γaction on its boundary ∂Γ. We claim that if π : Γ⟶H is a uniform lattice embedding into a locally compact group H, and Φ_{π} : H⟶QI (Γ) is a homomorphism (of abstract groups) satisfying (b) and (c) of Lemma 3.3, then

Theorem 3.5. The homomorphism

Ψ_{π} : H⟶^{Φ_{π}}QI (Γ)⟶^{β}Homeo (∂Γ)  (3.5)

is continuous with respect to the uniform convergence in Homeo (∂Γ). The composition homomorphism Ψ_{π} ∘ π : Γ⟶Homeo (∂Γ) is the standard Γ-action on its boundary. The image Ψ_{π}(H) is a locally compact subgroup of Homeo (∂Γ), containing Ψ_{π}(π(Γ)) as a uniform lattice. The kernel Ker(Ψ_{π}) is a compact normal subgroup of H, which has at most finite intersection with π(Γ).

Proof. Let us first recall some general facts about Gromov hyperbolic groups. Let d be some fixed left invariant proper metric d on Γ (say a word metric). The (ideal) boundary ∂Γ consists of equivalence classes of quasi-geodesic rays, which are by definitions quasi-isometric embeddings r : ℝ_{+} → (Γ, d), modulo the equivalence relation: r ∼ r′ if the distance D(r, r′) is finite. Recall that D(r, r′) is defined as

D(r, r′) := sup{d(r(t), r′(t)) | t ≥ 0}

This gives rise to the action of QI (Γ) on ∂Γ since for any quasi-geodesic ray r : ℝ_{+} → Γ and any quasi-isometry q of Γ, the composition q ∘ r : ℝ_{+} → Γ is a quasi-geodesic ray, and this definition respects the “bounded distance” equivalence relations on the rays and on the quasi-isometries. The resulting action of QI (Γ) on ∂Γ turns out to be continuous, and the corresponding homomorphism QI (Γ)⟶Homeo (∂Γ) was denoted by β.

In a Gromov hyperbolic group points ξ of the boundary ∂Γ (i.e. classes of quasigeodesic rays) can be represented by quasi-geodesic rays r : ℝ_{+} → Γ with r(0) = e_{Γ} and such that the quasi-geodesic constants are uniformly bounded, say by M_{0} and A_{0} (in general, such a representation of ξ by r_{ξ} with the above properties is not unique, but any two such rays are within finite distance which is uniformly bounded). The topology on ∂Γ admits a uniform structure given by neighborhoods U_{n} of the diagonal in ∂Γ×∂Γ, defined as follows: fix a sufficiently large constant C and define a decreasing sequence of neighborhoods U_{n} of the diagonal in ∂Γ × ∂Γ by declaring (ξ_{1}, ξ_{2}) to be in U_{n} if ξ_{1}, ξ_{2} can represented by (M_{0}, A_{0})-quasi-geodesic rays r_{1}, r_{2} : ℝ_{+} → Γ with r_{1}(0) = r_{2}(0) = e_{Γ} and

d(r_{1}(t), r_{2}(t)) ≤ C for t ∈ [0, n]

We shall also need the following general fact, often referred to as the Morse Lemma: given any (M, A)-quasi-geodesic ray r : ℝ_{+} → Γ, there exists a (M_{0}, A_{0})- quasi-geodesic ray r_{0} : ℝ_{+} → Γ with r_{0}(0) = e_{Γ} which is within bounded distance D(r, r_{0}) < ∞ from r, and moreover the distance D(r, r_{0}) can be bounded in terms of d(r(0), e_{Γ}), M, A (and M_{0}, A_{0}).

With these preliminary remarks, let us prove the continuity of the homomorphism

Ψ_{π} : H⟶^{Φ_{π}}QI (Γ)⟶^{β}Homeo (∂Γ)

Let M, A and B be constants as in Lemma 3.3.(c) and (d). For any n let

F_{n} = {γ ∈ Γ | d(γ, e) ≤ M_{0} · n + A_{0}}

It is a finite subset of Γ. By property (d) of Lemma 3.3 there is a neighborhood V_{n} of e_{H} in H such that

d(q_{h}(γ), γ) ≤ B  for γ ∈ F_{n}, h ∈ V  (3.6)

Take an arbitrary ξ ∈ ∂Γ represented by a (M_{0}, A_{0})-quasi-geodesic ray r_{ξ} : ℝ_{+} → Γ with r_{ξ}(0) = e_{Γ}. Note that r_{ξ}(t) ∈ F_{n} for t ∈ [0, n]. For any h ∈ H the map r′ = q_{h} ∘ r_{ξ} : ℝ_{+} → Γ is a quasi-geodesic ray (representing Ψ_{π}(h) · ξ ∈ ∂Γ) which has quasi-isometric constants (M_{1}, A_{1}) being bounded in terms of M, M_{0}, A, A_{0}.

For h ∈ V_{n} one also has d(r′(0), e_{Γ}) = d(q_{h}(e_{Γ}), e_{Γ}) ≤ B. For any such h ∈ V_{n} let r_{h·ξ} denote an (M_{0}, A_{0})-quasi-geodesic ray corresponding to r′ as in the “Morse Lemma” above. Then D(r_{h·ξ}, r′) is bounded in terms of M_{1}, A_{1} and B. Letting C = D(r_{h·ξ}, r′) + B, we obtain for all t ∈ [0, n] and h ∈ V_{n} the following estimate

d(r_{h·ξ}(t), r_{ξ}(t)) ≤ D(r_{h·ξ}, r′) + d(q_{h} ∘ r_{ξ}(t), r_{ξ}(t)) ≤ D(r_{h·ξ}, r′) + B = C

because r_{ξ}(t) ∈ F_{n} for t ∈ [0, n]. Note that the bound C is independent of n, and the above inequality proves that for all h ∈ V_{n}

(Ψ_{π}(h) · ξ, ξ) ∈ U_{n} (h ∈ V_{n}, ξ ∈ ∂Γ)

where U_{n} is defined using the uniform constant C, independent of n. This proves the continuity of Ψ_{π} : H⟶Homeo (∂Γ).

Since Φ_{π}∘π : Γ⟶QI (Γ) coincides with the natural homomorphism ρ : Γ⟶QI (Γ), the homomorphism

Γ⟶^{π}H⟶^{Φ_{π}}QI (Γ)⟶^{β}Homeo (∂Γ)

corresponds to the standard Γ-action on its boundary ∂Γ. The standard homomorphism Γ⟶Homeo (∂Γ) is known to have finite kernel and a discrete image, which we denote by Γ̄. Thus Ψ_{π}(Λ) forms a discrete subgroup of Ψ_{π}(H) and the kernel K = Ker(Ψ_{π}) ◁ H has at most finite intersection with Λ = π(Γ).

Now take a compact subset Q ⊂ H with Q Λ = H. By continuity, Ψ_{π}(Q) is compact and its translations by the discrete group Ψ_{π}(Λ) cover all of Ψ_{π}(H). This easily implies that (i) Ψ_{π}(H) is locally compact (in the topology of the uniform convergence), (ii) Ψ_{π}(Λ) is a uniform lattice in Ψ_{π}(H), and (iii) the kernel K = Ker(Ψ_{π}) is compact.

Returning to the situation where Γ is a uniform lattice in a simple Lie group G of rank one, denote by ∂X the visual boundary of the symmetric space X = G/K. Then ∂Γ is naturally identified with ∂X (as topological spaces with a Γ-action), and ∂Γ = ∂X is homeomorphic to the sphere S^{n−1} where n is the real dimension of X. The image Γ̄ ⊂ Homeo (∂X) of Γ is isomorphic to Γ/Z(Γ) ≅ Ad Γ. We shall also denote by L = Ψ_{π}(H) the locally compact subgroup of Homeo (∂X) constructed above. Our goal is to prove Claim 3.1.

There are several cases to be considered.

Case G ≃ SL_{2}(ℝ) (Theorem C). In this case the symmetric space X = G/K is the real hyperbolic plane H^{2} and ∂X is the circle S^{1}. We claim that, up to conjugation in Homeo (S^{1}), the locally compact group L = Ψ_{π}(H) ⊂ Homeo (S^{1}) is either PSL_{2}(ℝ) ≅ Ad G, or PGL_{2}(ℝ) ≅ Aut (Ad G), or is a a discrete group Γ_{∗} which forms a uniform lattice in PSL_{2}(ℝ) or in PGL_{2}(ℝ), containing Γ̄ ≅ Ad Γ as a finite index subgroup.

At this point one should point out that it is possible to describe quite explicitly all locally compact subgroups of Homeo (S^{1}). Here we shall outline the arguments required in our particular situation. It is convenient to consider the orientation preserving subgroup L_{+} = L∩Homeo_{+}(S^{1}), which has index at most two in L = Ψ_{π}(H).

The first main observation (pointed out to me by Etienne Ghys, whom I would like to thank) is that locally compact subgroups, such as L_{+}, of Homeo_{+}(S^{1}) have No Small Subgroups, i.e. one can find a neighborhood U of the identity in L_{+} which contains no non-trivial subgroups. It suffices to show that there are no small compact subgroups, and this follows from the following easy

Lemma 3.6. Any compact subgroup K ⊂ Homeo_{+}(S^{1}) is conjugate in Homeo_{+}(S^{1}) to a closed subgroup of rotations SO(2), and therefore K is either finite or coincides with SO(2).

Using Montgomery-Zippin’s results (see [MZ]) a locally compact group L_{+} with No Small Subgroups has a real connected Lie group as its connected component L_{+}^{0} of the identity.

Next recall that the action of Γ̄ on S^{1} is minimal and strongly proximal (this general fact applies to the actions of hyperbolic groups Γ on their boundaries ∂Γ, as well as for actions of lattices Γ on the Furstenberg’s boundary B(G) of the ambient semisimple Lie group G). Recall that minimality means that all orbits Γ̄ · x are dense in ∂Γ, while strong proximality means that Γ̄-orbit of any probability measure μ has Dirac measures in its closure with respect to weak topology. Since L_{+} contains Γ̄ its action on S^{1} is also minimal and strongly proximal.

Lemma 3.7. Let B be a metric compact space and L ⊂ Homeo (B) be a locally compact subgroup which acts minimally and strongly proximally on B. Then L has no non-trivial closed normal amenable subgroups.

Proof. Let N ◁ L be a closed normal subgroup, and let V_{N} ⊂ P(B) be the set of all N-invariant probability measures on B. Then V_{N} is a non-empty (because A is amenable), closed and convex subset of P(B) which is L-invariant. Strong proximality of the L-action implies that V_{N} contains some Dirac measures, and the minimality implies that all the Dirac measures {δ_{ξ}}_{ξ∈B} are in V_{N}, which means that N acts trivially on B.

Applying this to B = S^{1} and our group L_{+}, one concludes that the solvable part of the Levi decomposition of the Lie group L_{+}^{0} is trivial, so L_{+}^{0} is reductive, and moreover it is a semisimple Lie groups with trivial center and no compact factors. By Lemma 3.6 the maximal compact of L_{+}^{0} has to be isomorphic to SO(2) or to be trivial. Thus L_{+}^{0} is either isomorphic to PSL_{2}(ℝ) or is trivial.

One can show that in the first case, an isomorphism of L_{+}^{0} with PSL_{2}(ℝ) can be realized by a conjugation in Homeo_{+}(S^{1}) (the conjugation map coming from Lemma 3.6). Since L_{+}^{0} ≅ PSL_{2}(ℝ) is normal in L_{+}, the latter acts by automorphisms on PSL_{2}(ℝ), which has only one outer automorphism. This outer automorphism cannot be realized as an orientation preserving homeomorphism of S^{1}. This implies that L_{+} centralizers L_{+}^{0}, and one can check that the centralizer of PSL_{2}(ℝ) in Homeo_{+}(S^{1}) is trivial (in fact this is true already for the maximal compact subgroup SO(2) of PSL_{2}(ℝ)). Therefore L_{+} = L_{+}^{0} is conjugate in Homeo_{+}(S^{1}) to PSL_{2}(ℝ). Considering all of L in all of Homeo (S^{1}), one finds that if L^{0} is non-trivial, then L is conjugate to either Ψ_{π}(H) or to PGL_{2}(ℝ).

If L_{+}^{0} is trivial, then L_{+} and L = Ψ_{π}(H) are discrete subgroups of Homeo (S^{1}), because by Lemma 3.6 L_{+} has No Small Subgroups. Since L and L_{+} contain Γ̄ as a cocompact lattice, the index [L : Γ̄] is finite. Finally, we claim that L ⊂ Homeo (S^{1}) lies in fact in PSL_{2}(ℝ) and therefore forms a lattice there. This follows, for example, from Gabai’s result [Ga] stating that convergence groups are Fuchsian. Indeed, since Γ̄ is a convergence group, so is L.

This proves Claim 3.1 and Theorem C follows.

Case G ≃ SO(n, 1), n > 2. For this and the following cases of Γ being uniform in a rank one group G ≄ SL_{2}(ℝ), we shall use some known facts about the conformal structures on ∂X (following ideas which are going back to Mostow).

For G ≃ SO(n, 1), n > 2, the boundary ∂X ≅ S^{n−1} of the real hyperbolic space X = H^{n}, n > 2, carries a natural conformal structure (e.g. see [GP] 3.6). Any quasi-isometry class ϕ ∈ QI (X) defines a quasi-conformal map β(ϕ) of S^{n−1} with the quasi-conformal constant bounded in terms of the quasi-isometric constant M of ϕ. In fact β : QI (X)⟶QC(∂X) is an isomorphism (of abstract groups). The subgroup of isometries Isom (X) is mapped isomorphically by β onto the group C(∂X) of conformal maps. Hence, we have

Ψ_{π}(H) ⊆ QC(∂X) ⊂ Homeo (∂X), Ψ_{σ}(Aut (Ad G)) = C(∂X)

The fact that all {Φ_{π}(h)}_{h∈H} can be represented by quasi-isometries {q_{h}}_{h∈H} with a uniform bound on the quasi-isometric constant M ((c) of Lemma 3.3) means that Ψ_{π}(H) is uniformly quasi-conformal subgroup of QC(S^{n−1}). This group has an additional important property: its action on the space of distinct triples

∂^{3}X := {(ξ_{1}, ξ_{2}, ξ_{3}) ∈ ∂X × ∂X × ∂X | ξ_{i} ≠ ξ_{j}, i ≠ j}

is cocompact. This follows from the fact that already the action of Ψ_{σ}(Ad Γ) = Ψ_{π}(Λ) ⊂ Ψ_{π}(H) on the triple space is cocompact. Tukia [Tu] proved, that any uniformly quasi-conformal subgroup of QC(S^{n−1}), n > 2, whose action on the triple space is cocompact, is conjugate into C(S^{n−1}). Hence, there exists an f ∈ QC(∂X), so that

f^{−1} Ψ_{π}(H) f ⊂ C(∂X) = Ψ_{σ}(Aut (Ad G))

However, no conjugation is really needed in our situation. Since both Ψ_{π}(Λ) = Ψ_{σ}(Ad Γ) and f^{−1}Ψ_{π}(Λ)f are in C(∂X), Mostow’s rigidity implies that for some g ∈ C(∂X), one has

(g ∘ f)^{−1} Ψ_{π}(λ) (g ∘ f) = Ψ(λ), (λ ∈ Λ)

In other words, g ∘ f is a Λ-equivariant continuous (actually quasi-conformal) map of ∂X to itself. However, the Λ-action (i.e. the Ad Γ-action) on ∂X is known to be minimal and proximal. It is easy to see that such actions have no non-trivial equivariant continuous maps. Hence f = g^{−1} ∈ C(∂X), and one has Ψ_{π}(H) ⊆ C(∂X) = Ψ_{σ}(Aut (Ad G)). This proves Claim 3.1 for uniform lattices Γ in G ≃ SO(n, 1), n > 2.

Case G ≃ SU(n, 1), n > 1. Here the symmetric space X is the complex hyperbolic spaces CH^{n} (of real dimension 2n), and the appropriate conformal structure on ∂X corresponds to the Carnot-Caratheodory metric. As in the previous case, the group Ψ_{π}(H) is uniformly quasi-conformal and its action on the triple space is cocompact. Following Tukia, Chow [Ch] proved that any such subgroup is conjugate into C(∂X). As in the previous case, the fact that Ψ_{π}(Λ) is already in C(∂X), together with Mostow Rigidity, imply that no conjugation is needed. Thus

Ψ_{π}(H) ⊆ C(∂X) = Ψ_{σ}(Aut (Ad G))

and Claim 3.1 is proved.

Case G ≃ Sp(n, 1) or F_{4}^{−20}. The symmetric spaces in these cases are the quaternionic hyperbolic space X = QH^{n} (corresponding to G ≃ Sp(n, 1)) and the Cayley plane X = CaH^{2} (corresponding to G ≃ F_{4}^{−20}). These spaces are very rigid: Pansu [Pa] proved that for these spaces QC(∂X) = C(∂X), while the latter again coincides with Ψ_{σ}(Aut (Ad G)), and Claim 3.1 follows.

Remark 3.8. Marc Bourdon suggested the following elegant argument for the above cases of rank-one G ≄ PSL_{2}(ℝ). Let ν_{0} denote the K-invariant measure on the boundary ∂X. It is well known that there is a function (closely related to the Busemann cocycle) f : ∂^{2}X → ℝ_{+} such that the infinite Radon measure dm_{0}(x, y) = dν_{0}(x)dν_{0}(y)/f(x, y) on the spaces of distinct pairs ∂^{2}X of points on the boundary is invariant and ergodic for the diagonal action of Γ̄. Moreover, the conformal group can be identified in Homeo (∂X) as the stabilizer of m_{0} (see Sullivan [Su] for SO(n, 1)):

Ψ_{σ}(G) = C(∂X) = {h ∈ Homeo (∂X) | (h × h)_{∗}m_{0} = m_{0}}  (3.7)

A fundamental step in Mostow’s proof of strong rigidity implies that quasi-conformal mappings of ∂X preserve the measure class of ν_{0}. Hence m_{0} is quasi-invariant for the diagonal Ψ_{π}(H)-action on ∂^{2}X and therefore the Radon measure

m_{1} = ∫_{H/Γ}(Ψ_{π}(h) × Ψ_{π}(h))_{∗}m_{0} dm_{H/Γ}(h)

is Ψ_{π}(H)-invariant and is in the same measure class as m_{0}. Since m_{0} is Γ̄-ergodic, m_{1} = m_{0} and (3.7) yields Ψ_{π}(H) ⊆ Ψ_{σ}(G).

Hence Claim 3.1 is proven and the proof of Theorem B can be completed exactly as in the Subsections 2.3 and 2.4 (the compactness of the kernel Ker(Φ : H⟶Aut (Ad G) in these cases can be deduced from the last assertion of Theorem 3.5).

3.4. Γ is a uniform irreducible lattice in a semisimple G of higher rank. For an irreducible higher rank symmetric space of non compact type X (G is simple of higher rank), it was proved by Kleiner and Leeb [KL], and independently by Eskin and Farb [EF], that QI (X) ≅ Isom (X). Hence, one has a homomorphism

Φ_{π} : H⟶QI (Γ) ≅ QI (X) ≅ Isom (X) ≅ Aut (Ad G)

In the case of a semisimple G, the symmetric space X = X_{1}×⋯×X_{r} is a product of symmetric spaces. For this case Kleiner and Leeb [KL] proved that QI (X) is equivalent (up to permutation of the factors) to QI (X_{1})×⋯×QI (X_{r}). Applying the above arguments to each of the factors one realizes that Ψ_{π}(H) is contained, up to finite index, in Homeo (∂X_{1})×⋯×Homeo (∂X_{r}), and in fact Ψ_{π}(H) ⊆ Ψ_{σ}(Aut (Ad G)) (without finite index reduction in this statement), so that Φ : Ψ_{σ}^{−1} ∘ Ψ_{π} : H⟶Aut (Ad G) is a continuous homomorphism satisfying (2.1). We should remark that the continuity of Ψ_{π} follows from the fact that the above factorization of QI (Π_{1}^{r}X_{i}) into the product Π_{1}^{r}QI (X_{i}) is effective in terms of the quasi-isometric constants (so an analogue of Theorem 3.5 holds). Special care should be taken of the case where one/some of the factors X_{i} are hyperbolic planes H^{2} (corresponding to PSL_{2}(ℝ) factors of G). In this case L = Ψ_{π}(H) is a locally compact subgroup of Π_{1}^{r}Homeo (∂X_{i}) which contains copy/ies of Homeo (S^{1}), and one can verify that the arguments, similar to the case of a single Homeo (S^{1}), still apply (in particular such L has no small subgroups).

Remark 3.9. The case of lattices Γ in higher rank semi-simple Lie groups should probably be treated in the measure-theoretical framework of Theorem A, by resolving the difficulty pointed out in Remark 2.2.

3.5. Γ is non-uniform in G ≄ SL_{2}(ℝ). Non-uniform irreducible lattices Γ in semisimple Lie groups G ≄ SL_{2}(ℝ) have only “algebraic” quasi-isometries, i.e. the quasiisometry group QI (Γ) coincides with the commensurator group Comm (Γ) of Γ in Isom (X) ≅ Aut (Ad G), defined by:

Comm (Γ) := {g ∈ Aut (Ad G) | [Γ : g^{−1}Γg ∩ Γ] < ∞}

This remarkable fact was first discovered by Schwartz [S1] for non-uniform lattices in rank one simple Lie groups; and then proved for some higher rank cases by Farb-Schwartz [FS] and Schwartz [S2]. Using different ideas Eskin [Es] proved the above result when G has no rank-one factors. The combination of these methods enables to deduce the result in its full generality, as it is described by Farb ([Fa], pp. 710–711).

The commensurator Comm (Γ) is a countable (dense) subgroup of Aut (Ad G). Clearly, Φ_{π}^{−1}({e}) is a measurable subset of H with a finite but positive Haar measure:

0 < c := m_{H}(Φ_{π}^{−1}({e})) ≤ m_{H}(H/Λ) < ∞

Hence, the push forward measure (Φ_{π}) ∗ m_{H} on Aut (Ad G) is an atomic measure, equally distributed on some subgroup Γ_{∗} ⊃ Ad Γ. Moreover, Ad Γ has finite index in Γ_{∗}, for

c · [Γ_{∗} : Ad Γ] = m_{H}(H/Γ) < ∞

Hence Γ_{∗} is a lattice in Aut (Ad G), and taking H_{0} := Φ_{π}^{−1}(Ad Γ) we conclude the proof as in Subsection 2.4, Case (2).

The proof of Theorem B is now completed.

3.6. An Exotic Example. In this section we shall describe an example which shows that in Theorem C case (1) one cannot replace the “almost direct product” structure of H over a compact abelian group C, by a similar one over a finite abelian C.

Let Γ be a surface group of a compact Riemann surface of genus g. As an abstract group Γ has the presentation

Γ = ⟨a_{1}, …, a_{g}, b_{1}, …, b_{g} | [a_{1}, b_{1}][a_{2}, b_{2}] ⋯ [a_{g}, b_{g}] = 1⟩

Such Γ can be embedded as a uniform lattice in G = SL_{2}(ℝ). Let G̃ be the universal covering of G

1⟶Z⟶G̃⟶^{p}G⟶1

with Z ≅ ℤ. The preimage Γ̃ = p^{−1}(Γ) is a non-trivial central extension of Γ by Z, which can be presented as

Γ̃ = ⟨a_{1}, …, a_{g}, b_{1}, …, b_{g}, z | [a_{i}, z] = [b_{i}, z] = e, [a_{1}, b_{1}][a_{2}, b_{2}] ⋯ [a_{g}, b_{g}] = z⟩

where z generates the center Z. The crucial observation is that Γ̃ is linear. Indeed, if ρ is a the Γ̃-representation in the Heisenberg group

ρ(a_{i}) = ( \begin{matrix}1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) ,  ρ(b_{i}) = ( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{matrix} ) ,  ρ(z) = ( \begin{matrix}1 & 0 & g \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} )

then Ad × ρ is a faithful linear Γ̃-representation. (I would like to thank Gregory Soifer for pointing this out to me).

Hence Γ̃ is residually finite. Let K′ be a profinite completion of Γ̃. The center \overline{Z} of K′ is a profinite completion of Z ≅ ℤ.

Now let G′ be defined as the quotient of the direct product \overline{Z} × G̃ by the diagonal embedding of Z. This embedding is discrete (due to the G̃-coordinate), so that G′ becomes a locally compact group, which contains G̃ as a dense subgroup. Hence G′ is connected, although it is neither locally connected, nor path-connected. The center of G′ is a compact (profinite) abelian group, isomorphic to \overline{Z}. Observe, that Γ̃ embeds in G′, via Γ̃ ⊂ G̃ ⊂ G′, however its image is no longer discrete.

Next, consider the locally compact group H := (G′ × K′)/\overline{Z}, where \overline{Z} is embedded diagonally in the centers of G′ and K′. Denoting by π′ the diagonal embedding of Γ̃ into G′ × K′, one observes that the homomorphism

Γ̃⟶^{π′}G′ × K′⟶H

has Z as its kernel, and therefore, factors through Γ as

π′ : Γ̃⟶^{p}Γ⟶^{π}H

where π is injective. We claim that π(Γ) forms a lattice in H. Indeed, H contains two closed commuting subgroups, isomorphic to G′ and K′ (these are the projections of G′ × {e_{K′}} and {e_{G′}} × K′, respectively), which we shall again denote by G′ and K′, and G′, K′ ◁ H intersect along their common center, which is isomorphic to \overline{Z}. Note that if Φ : H⟶H/K′ ≅ G denotes the projection, then

Γ⟶^{π}H⟶^{Φ}G

coincides with the embedding Γ ⊂ G. Since K′ = Ker(Φ : H⟶G) is compact, we conclude that π is an embedding of Γ as a uniform lattice in H.

Any connected subgroup of H, which projects onto G, has to contain G̃ ⊂ G′, and any closed connected one contains G′, which has a compact, but not finite, abelian center \overline{Z}.

Remark 3.10. The above construction does not work for other simple Lie groups G, which have universal covering with an infinite center. It is known that for such G, there does not exist a residually finite Γ′ ⊂ Γ̃ ⊂ G̃ which projects onto Γ ⊂ G. However, it is not clear to the author whether a residually finite Γ′ projecting on Γ can be found in G′ = G̃ × \overline{Z(G)}/Z(G).

Remark 3.11. A construction similar to (but somewhat simpler than) the above one gives an embedding of a surface group Γ in a locally compact group H ≅ G′ × K′/Z, where G′ is a simple Lie group with a finite center Z, with G′ being locally isomorphic but not isomorphic to G.

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Department of Mathematics (m/c 249), University of Illinois at Chicago, 851 South Morgan Street, Chicago, IL 60607-7045

  E-mail address: furman@math.uic.edu