The goal of these lectures is to state and prove Richard Schwartz’s theorem on the quasi-isometry rigidity of nonuniform lattices in real hyperbolic space \( \mathbb{H}^n \) (for \( n \ge 3 \)).

Basic definitions

Coarsely Lipschitz applications Let \( X, Y \) be metric spaces. An map \( f : X \to Y \) is coarsely \( (L, A) \)-Lipschitz if for every \( x_1, x_2 \in X \) we have
\[
d(f(x_1), f(x_2)) \le Ld(x_1, x_2) + A.
\]

Coarse inverse Let \( f : X \to Y \) be an map ; an map \( \bar f : Y \to X \) is called an coarse inverse of \( f \) if both \( f \bar f \) and \( \bar f f \) are at bounded distance from the identity. That is, there exists \( C \) such that
\[
\forall x \in X : d(x, \bar f f(x)) \le C, \: \forall y \in Y : d(y, f\bar f(y)) \le C.
\]

Quasi isometry A map \( f : X \to Y \) is said to be a quasi-isometry if it is coarsely Lipschitz and has a coarse inverse which is as well.

The Milnor–Schwarz lemma is the statement that a finitely generated group \( \Gamma \) (with the word metric associated to any finite generated set) acting cocompactly on a geodesic metric space \( X \) is quasi-isometric to \( X \).

Virtual isomorphism This is the equivalence relation on finitely generated groups generated by \( \Gamma_1 \sim \Gamma_2 \) whenever there exists a finite group \( F_1 \) and a short exact sequence
\[
1 \to F_1 \to \Gamma_2 \to \Gamma_1 \to 1
\]
or \( \Gamma_1 \) is a finite-index subgroup in \( \Gamma_2 \). Clearly virtual isomorphism implies quasi-isometry. Note that two commensurable groups (meaning they have isomorphic finite-index subgroups) are virtually isomorphic but the converse is not true in general.

Quasi-isometric rigidity A finitely generated group \( \Gamma \) is said to be quasi-isometrically rigid if any finitely generated group \( \Gamma’ \) which is quasi-isometric to \( \Gamma \) is virtually isomorphic to \( \Gamma \).

\( \mathbb{H}^n \) is \( n \)-dimensional hyperbolic space, which will be represented by the upper half space model \( x \in \mathbb{R}^n, x_n > 0 \) with te Riemannian metric \( dx^2/x_n^2 \). The isometry group \( G = \mathrm{Isom}(\mathbb{H}^n) \) is isomorphic to \( \mathrm{PO}(n, 1) \).

Definition: A discrete subgroup \( \Gamma \subset G \) is called a lattice if the volume of \( \Gamma \backslash \mathbb{H}^n \) is finite. It is said to be uniform if the quotient is compact, and non-uniform otherwise.

Schwartz’s theorem is the following statement.

Theorem: Nonuniform lattices in \( \mathbb{H}^n \) are quasi-isometrically rigid.

Since there are noncommensurable cocompact lattices in \( \mathbb{H}^n \) QI-rigidity does not hold for these. However, a theorem of Sullivan–Tukia states they are QI-rigid as a class, that is a group which is quasi-isometric to a cocompact lattice in \( \mathbb{H}^n \) must be virtually isomorphic to an (a priori different) lattice in \( \mathbb{H}^n \).

A more concrete definition of nonuniform lattices in hyperbolic space

Horospheres and balls In the upper half-space model the boundary at infinity \( \partial_\infty \mathbb{H}^n \) is identified with \( \mathbb{R}^{n-1} \times \{ 0 \} \cup \{ \infty \}\). If \( \lambda \in \mathbb{R}^{n-1} \times \{ 0 \} \) an horoball of center \( \lambda \) is an Euclidean ball in the upper half-space which is tangent to \( \mathbb{R}^{n-1} \times \{ 0 \} \) at \( \lambda \) ; if \( \lambda = \infty \) it is an half space of the form \( \{ x : x_n \ge a \} \) for some \( a > 0 \). An horosphere is the boundary of an horoball.

Truncated hyperbolic spaces Let \( \mathcal B \) be a collection of pairwise disjoint horoballs in \( \mathbb{H}^n \). Then
\[
\Omega = \mathbb{H}^n \setminus \bigcup_{B \in \mathcal B} B
\]
is a truncated hyperbolic space. The set of centers of horoballs in \( \mathcal B \) will usually be denoted by \( \Lambda \).

Theorem (Garland–Raghunathan): A discrete subgroup \( \Gamma \subset G \) is a lattice if and only if there exists a truncated hyperbolic space \( \Omega \subset \mathbb{H}^n \) such that \( \Gamma \) preserves \( \Omega \) and the quotient \( \Gamma \backslash \Omega \) is compact.

The lattice is uniform if and only if \( \Omega \) is equal to hyperbolic space \( \mathbb{H}^n \).

The following statement implies Schwartz’s theorem for lattices in \( \mathbb{H}^n \).

Theorem: Suppose that \( n \ge 3 \). Let \( \Gamma, \Gamma’ \) be two nonuniform lattices in \( \mathbb{H}^n \) acting cocompactly on truncated spaces \( \Omega, \Omega’ \). If there exists a quasi-isometry \( f : \Omega \to \Omega’ \) then \( f \) is at finite distance from (the restriction of) an isometry of \( \mathbb{H}^n \).

Remarks

1. The rest of the statement (which is similar to the one in the cocompact case) is somewhat simpler to prove.
2. This statement is false when \( n = 2 \), for example lifting a diffeomorphism between two noncompact noninsometric surfaces gives a counterexample. However, nonuniform lattices in \( \mathrm{PO}(2, 1) \) are virtually free groups and these are quasi-isometrically rigid as a consequence of atheorem of Stallings.

Examples of nonuniform lattices

1. If \( n = 2 \), then \( \mathrm{PO}(2, 1)^0 \cong \mathrm{PSL}(2, \mathbb{R}) \) and \( \mathrm{PSL}(2, \mathbb{Z}) \) is a nonuniform lattice. The set of center of horoballs is \( \mathbb{Q} \cup \{\infty\} \).
2. If \( n = 3 \) then \( \mathrm{PO}(3, 1)^0 \cong \mathrm{PSL}(2, \mathbb{C}) \). If \( D > 0 \) is a square-free integer then \( \mathrm{PSL}(2, \mathbb{Z}[\sqrt{-D}]) \) is a nonuniform lattice. The center of horospheres are \( \mathbb{Q}(\sqrt{-D}) \cup \{\infty\} \). These groups are called Bianchi groups. They are pairwise noncommensurable.
3. For \( n \ge 4 \) there are so-called arithmetic lattices in \( \mathrm{PO}(n, 1) \) which are constructed from isotropic quadratic forms over \( QQ \), for example \( \mathrm{PO}(n, 1)(\mathbb{Z}) \) is associated to the quadratoc form \( x_1^2 + \cdots + x_n^2 – x_{n+1}^2 \). On the other hand not much is known about the general global structure of finite volume hyperbolic manifolds in thses dimensions.

Other QI-rigidity results for lattices in symmetric spaces

1. For other symmetric spaces of real rank 1 (complex and quaternionic hyperbolic spaces, associated to the Lie groups \( \mathrm{SU}(n, 1), n \ge 2 \) and \( \mathrm{Sp}(n, 1), n \ge 2 \), and the octonionic hyperbolic plane associated to the exceptional group \( F_4^{-20} \)) QI-rigidity was also proven by Schwartz, an essential ingredient being a theorem of Pansu.
2. For lattices in products of rank 1 spaces QI-rigidity was proven by Farb and Schwartz.
3. For lattices in symmetric spaces without rank 1 factors QI rigidity was proven by Eskin.
4. For lattices in non-Archimedean groups there is work of K. Wortman.
5. For some non-irreducible lattices it is not known whether QI-rigidity holds or not, for example \( \mathrm{SL}(2, _ZZ) \times \mathrm{SL}(2, \mathbb{Z}) \).

An outline of Schwartz’s proof

For the whole proof \( \Omega, \Omega’ \) are truncated hyperbolic spaces acted upon cocompactly by nonuniform lattices \( \Gamma, \Gamma’ \) and \( f : \Omega \to \Omega’ \) is a quasi-isometry.

First step

The proof begins by proving that \( f \) coarsely preserves horospheres in \( \partial \Omega \), that is for every component \( \Sigma \subset \partial\Omega \) there exists another \( \Sigma’ \subset \partial\Omega’ \) such that \( f(\Sigma) \) is at finite Hausdorff distance from \( \Sigma’ \).

Note that already at this step the proof breaks when \( n= 2 \). There exists quasi-isometries between truncated hyperbolic planes sendind an horocycle to a geodesic, because they are quasi-isometric to trees and tree automorphisms do not respect horocycles.

If is not hard to go from the result of this step to supposing that in fact \( f(\partial \Omega) = \partial\Omega’ \), which will always be assumed in the sequel.

Second step

In this step it is proven that it is possible to extend \( f \) to a quasi-isometry \( \tilde f \) of \( \mathbb{H}^n \) to itself.

The rest of the proof consists in showing that there exists an isometry \( \alpha \) of \( \mathbb{H}^n \) at finite distance of \( \tilde f \).

Third step

It is possible to extend \( \tilde f \) to an homeomorphism \( \partial\tilde f \) of \( \partial_\infty \mathbb{H}^n \). If \( n \ge 3 \) then this homeomorphism is quasiconformal (this is a result due to Efremovich–Tikhomirova and independently Mostow). Then it is possible to use the following theorem.

Theorem (Gehring): A quasiconformal homeomorphism is differentiable at almost every point. If its differential is conformal at almowt every point then it is a Möbius transformation.

Remark: In higher rank it is possible to use additional structure on the boundary to prove QI-rigidity from here. In rank one it is necessary to use hard analysis instead.

Fourth step

In the sequel \( h = \partial\tilde f \). This step and the next are dedicated to analysing the derivative \( Dh_x \) at points where it exists. We may assume that \( x = 0 \) and \( h(0) = 0 \). Then if \( A = Dh_0 \) we have
\[
Dh_0(v) = \lim_{t \to 0} \left( t^{-1}h(tv) \right) = \lim_{t \to 0} \left( h_{-t} (h(h_t(v))) \right)
\]
where \( h_t \) is the Möbius transformation \( x \mapsto tx \). This is used to prove that there are truncated spaces \( \Omega_\infty, \Omega_\infty’ \) and lattices \( \Gamma_\infty, \Gamma_\infty’ \) (conjugated to the original lattices), such that the sets of centers of horospheres \( \Lambda_\infty, \Lambda_\infty’ \) satisfy \( A\Lambda_\infty = \Lambda_\infty’ \).

Fifth step

In this step it is assumed that there are horospheres \( \Sigma_\infty \subset \partial \partial\Omega, \Sigma_\infty’ \subset \partial \partial\Omega’ \) with center \( \infty \). Let \( J \) be the inversion in the unit sphere of \( \mathbb{R}^n \) and \( B = J\tilde A J \) (where \( \tilde A \) is the extension of \( A : \mathbb{R}^{n-1} \to \mathbb{R}^{n-1} \) to a linear automorphism of \( \mathbb{R}^n \). Then assuming that \( A \not\in \mathbb{R}^\times \mathrm O(n-1) \) it is shown that
\[
d(\Sigma_\infty’, B\Sigma_i) \to \underset{i \to +\infty}{\to} +\infty
\]
where \( \Sigma_i \) are the horospheres in \( \partial\Omega \) at minimal distance of \( \Sigma_\infty \). This is a contradiction and thus \( A \) must be a similarity.

Sixth step

By Gehring’s theorem and the previous step \( h \) is Möbius, and thus extends to an isometry \( \alpha \) of \( \mathbb{H}^n \) such that \( \tilde f \) and \( \alpha \) are at bounded distance. It remains to show that \( \Gamma’ \) and \( \alpha\Gamma\alpha^{-1} \) are commensurable. This finishes the proof in the case when \( \Gamma \) and \( \Gamma’ \) are lattices

Seventh step

This step deals with the case when a group \( \Gamma’ \) is quasi-isometric to a truncated space \( \Omega \) on which a lattice \( \Gamma \) acts cocompactly.

Quasi-actions Let \( f : \Gamma’ \to \Omega \) be a quasi-isometry. Then we have a quasi-morphism \( \phi \) from \( \Gamma’ \) to the semigroup \( \mathrm{QI}(\Omega) \) of quasi-isometries of \( \Omega \) given by \( \phi(\gamma) = f\gamma \bar f \) where \( \bar f \) is a quasi-inverse of \( f \).

Steps 1–5 above can be applied to each \( \phi(\gamma) \) and in this way we obtain a quasi-morphism \( \phi^* \) from \( \Gamma’ \) to \( \mathrm{Isom}(\Omega) \). Because two isometries which are at finite distance must be equal \( \phi^* \) is in fact a morphism.

Because \( \phi^* \) is at finite distance from the quasi-isometry \( f \) the kernel \( \ker(\phi^*) \) must be finite. On the other hand the group of isometries of \( \mathbb{H}^n \) which preserve \( \Omega \) is discrete and thus must be commensurable to the lattice \( \Gamma \). It follows that \( \phi^*(\Gamma’) \) is commensurable to \( \Gamma \). These two facts together show that \( \Gamma’ \) is virtually isomorphic to \( \Gamma \).

Details for Step 1

Let \( \Sigma \) be an horosphere in \( \partial\Omega \). We want to find an horosphere \( \Sigma’ \subset \partial\Omega’ \) such that \( f(\Sigma) \) stays within bounded distance of \( \Sigma’ \).

Note that \( \Omega, \Omega’ \) are CAT(0)-spaces, and if \( n \ge 3 \) the Tits metric on the boundary \( \partial_\infty\Omega \) has two types of connected components: isolated points which correspond to geodesic rays not meeting any boundary component, and the boundary of the flat horosphers, which are \( n-2 \)-spheres. Thus \( \partial f \) sends the \( n-2 \)-spheres in \( \partial_\infty\Omega \) to those in \( \partial_\infty\Omega’ \). By using the filling techniques in Urs Lang’s lectures it is then possible to deduce the result.

Proof using coarse topology Suppose now that \( f \) is continuous. Any hypersurface which does not stay within bounded distance of an horosphere separates \( \Omega’ \), which implies that \( f(\Omega) \) is at bounded distance of an horosphere.

If \( n \ge 3 \), for any \( R > 0 \) the \( R \)-neighbourhood \( N_R(\Sigma) \) does not separate \( \Omega \). This makes possible a proof along the lines of the above in a coarse setting. The main ingredient if this is a version of the higher-dimensional Jordan curve theorem. This can be established using a coarse version of homology and Alexander duality.

Details for Step 2

We assume that \( f(\partial\Omega) = \partial\Omega’ \). The most natural way to extend \( f \) inside the horoballs is as follows. Let \( x \in \partial \Omega \) lie on an horosphere \( \Sigma \). There is a unique geodesic ray \( \rho \) from \( x \) to the center of \( \Sigma \), and these rays sweep all of the horoball as \( x \) moves along \( \Sigma \). Likewise, there is a unique geodesic ray from \( f(x) \) to the center of the horosphere \( \Sigma’ \) on which it lies. We can define \( \tilde f \) on \( \rho \) by sending it isometrically onto \( \rho’ \). It is then possible that \( \tilde f \) is a quasi-isometry by elementary hyperbolic geometry arguements.

Details for Step 3

Let \( \xi \in \partial_\infty\mathbb{H}^n \) be represented by a geodesic ray \( \rho \). Then the extension \( h = \partial\tilde f \) is defined by setting \( \rho’ = \tilde f \circ \rho \), which is a geodesic ray by the Morse lemma, and hence defines a point \( h(\xi) \in \partial_\infty\mathbb{H}^n \). We want to show that \( h \) is quasiconformal. For this, by composing with Möbius transformations we may suppose that \( h(\infty) = \infty \).

For this we need to preove that if \( c \) is a (euclidean) circle of center \( \xi \) (we assume \( \xi, \xi’ \in \mathbb{R}^{n-1} \)) then there exists Euclidean circles \( c’, c » \) with center \( \xi’ \) and radii \( R > r \) such that \( h(c) \) is contained in the annulus with boundary \( c’ \cup c » \) and \( R/r < K \), with \( K \) depending only on the quasi-isometry constants of \( \tilde f \).

For this we observe that \( c \) is the preimage under the orthogonal projection \( \pi \) to the geodesic \( [\xi, \infty] \) of a point \( x \in [\xi, \infty] \). Because \( \tilde f \) is a quasi-isometry, if \( \pi’ \) is orthogonal projection to \( [\xi’, \infty] \) then \( \tilde f \circ \pi \) and \( \pi’ \circ \tilde f \) are at a bounded distance. Thus \( \tilde f(c) \) is contained in the preimage of an interval of diameter \( \log(K) \) around \( \pi'(\tilde f(x)) \) where \( K \) depends only on \( \tilde f \). But this preimage is exactly an annulus satisfying the desired condition.

This proves that \( h \) is quasiconformal. By facts from analysis it follows that it is differentiable with invertible derivative at almost every point.

Detail for Step 4

To simplify assume that \( \xi = \xi’ = 0 \). We may choose \( y \in \mathbb{H}^n \) such that the geodesic ray \( [y, 0] \) intersects \( \partial\Omega \) in inifinitely many points \( y_1, \ldots, y_i, \ldots \) (this is the case for almost every \( y \) because of the erfodicity of the geodesic flow on \( \Gamma \backslash \mathbb{H}^n \)). We may modify the data by an osmetry to assume that \( y \in [0, \infty] \) so that there exists a sequence of positive real numbers \( t_i \to 0 \) such that \( y_i = t_i y \).

Let \( T_i : x \mapsto t_i x \), this is an isometry of \( \mathbb{H}^n \). Moreover we have:
\[
A := Dh_0(x) = \lim_{i \to +\infty} \left( \frac 1{t_i}h(t_i x) \right) = \lim_{i \to +\infty} T_i^{-1}hT_i(x).
\]
Lett \( \tilde f_i = T_i^{-1}\tilde f T_i \), this is a quasi-isometry with the same constants as \( \tilde f \).

Cocompacity of the action of \( \Gamma \) on \( \Omega \) allows us to choose \( \gamma_i \in \Gamma \) such that \( d(y_i, \gamma_i y) \le C \) (where \( C \) depends only on \( \Gamma, \Omega \)). Likewise there are \( \gamma_i’ \in \Gamma’ \) tels que \( d(y_i’, \gamma_i’ y’) \le C \) where \( y_i’ = \tilde f(y_i) \). It follows that the sequences \( k_i = T_i^{-1}\gamma_i, k_i’ = T_i^{-1}\gamma_i’ \) are bounded in \( \mathrm{Isom}(\mathbb{H}^n) \) (because they move \( y \) by a bounded amount) and we can extract subsequences converging to \( k_\infty, k_\infty’ \). Likewise, \( \gamma_i’ \tilde f_i \gamma_i \) is a boiunded sequence of quasi-isometries and it is at bounded distance of \( \tilde f_i \) and hence we can extract a subsequence of the latter converging towards a quasi-isometry \( \tilde f_\infty \).

Finally we define \( \Omega_\infty = k_\infty \) and \( \Gamma_\infty = k_\infty\Gamma k_\infty^{-1} \), and likewise \( \Omega_\infty’, \Gamma_\infty’ \). Then the action of \( \Gamma_\infty, \Gamma_\infty’ \) on \( \Omega_\infty, \Omega_\infty’ \) are cocompact and \( \tilde f_\infty \) is a quasi-isometry, with boundary extension \( \partial\tilde f_\infty = A \).

Detail for Step 5

According to Step 4 we may assume that the boundary extension \( A = \partial\tilde f \) is a linear map (we drop all \( \infty \) indices from the previous step to simplify notation). We want to prove that it is a similarity. To do this let \( J \) be the inversion around the unit sphere in \( \mathbb{R}^{n-1} \). Then by linear algebra \( A \) is a similarity if only if \( A’ := JAJ \) is still a linear map. In what follows we suppose this is not the case and we derive a contradiction.

We assumed that \( \infty \) is the center of an horoball \( B_\infty \subset \partial\Omega \), let \( L_\infty \) be its stabiliser in \( \Gamma \), which is a Euclidean lattice in \( \mathbb{R}^{n-1} \). Likewise \( L_\infty’ \) is the stabiliser of infinity in \( \Gamma’ \). Then we may prove that there are sequences of translations \( \tau_k \in L_\infty \) and \( \tau_k’ \in L_\infty’ \) such that both tend to infinity in \( \mathbb{R}^{n-1} \) and \( A_k’ := \tau_k’ A’ \tau_k \) converges to an affine map on \( \mathbb{R}^{n-1} \).

Because we assumed that \( A’ \) is not linear we must have that \( A_k’ \) is not stationary. Thus we may find \( \lambda \in \Lambda \) and \( \lambda’ \in \Lambda’ \) such that \( \lambda_k := A_k’\lambda \to \lambda’ \) but \( \lambda_k \not= \lambda’ \) for all \( k \). Let \( \Sigma_k’ \) be the horosphere in \( \partial\Omega \) centered at \( \lambda_k \) ; then we must have \( d(\Sigma_\infty, \Sigma_k ) \to +\infty \) as \( k \to +\infty \). On the other hand, because \( A \) extends the quasi-isometry \( \tilde f \) and \( \tau_k, \tau_k’ \) preserve \( d(\Sigma_\infty, \cdot) \) and \( d(\Sigma_\infty’, \cdot) \) this is not possible.