Ingredients and consequences of quasi-isometric rigidity of lattices in certain solvable Lie groups (Tullia Dymarz)

This is a transcript of my notes from Tullia Dymarz’s lectures. They may not always accurately reflect the content of the lectures, especially in places I did not put down everything that was said, and some comments or details might be mine.

  1. First lecture
  2. Second lecture
  3. Third lecture
  4. Fourth lecture
  5. Fifth lecture

Lecture I

General Outline and references

This series of lectures will be concerned with the following topics.

  1. Quasi-isometric rigidity of uniform lattices in \( \mathbb{H}^{n+1} \) for \( n \ge 2 \) (Tukia’s theorem). The reference here is Drutu–Kapovich’s book Geometric group theory (available here).
  2. QI-rigidity of lattices in certain solvable Lie groups, using generalisations of Tukia’s theorem. This is work of Dymarz and Xie, detailed in their paper Day’s fixed point theorem, Group cohomology and Quasi-isometric rigidity.
  3. Comparison between bi-Lipschitz and quasi-isometric equivalence for discrete groups, following Dymarz’s paper Bilipschitz equivalence is not equivalent to quasi-isometric equivalence for finitely generated groups, which shows that certain lamplighter groups are quasi isometric to each other but not bi-Lipshitz.

Preliminary material on quasi-isometries

Definition of quasi-isometric rigidity

Let \( X \) be a metric space. We say that it is quasi-isometrically rigid (« QI-rigid ») if for any finitely generated group \( \Gamma \) which is quasi-isometric to \( X \) there exists a morphism \( \Gamma \to \mathrm{Isom}(X) \) with finite kernel, and with image acting properly discontinuously and cocompactly.

The quasi-isometry group of a space and quasi-actions

Recall the definition of a \( (K, C) \)-quasi-isometry of \( X \) from Kapovich’s lectures: it is a map \( X \to X \) which is a \( (K, C) \)-quasi-isometruc embedding, that is
\[
\frac 1 K d(x, x’) – C \le d(f(x), f(x’)) \le Kd(x, x’) + C
\]
and in addition has a quasi-inverse \( \bar f \) which is also a \( (K, C) \)-quasi isometry. This means that there exists \( D \) such that \( d(\bar f f(x), x) \le D \) and \( d(f\bar f(x), x) \le D \) for all \( x \in X \). We will also say that \( \bar f f \) (and \( f \bar f \)) are at bounded distance from the identity.

The set of quasi-isometries of \( X \) is not a group because maps in there often fail to have an inverse. Let us introduce the equivalence relation \( \sim \) on the self-maps of \( X \), defined by \( f \sim g \) if and only if \( f \) and \( g \) are at bounded distance of each other, i.e. \( \exists D : \forall x \in X d(f(x), g(x)) \le D \). The quasi-isometry group of \( X \) is then defined to be
\[
\mathrm{QI}(X) = \{ \text{quasi-isometries of } X \} / \sim
\]
which is a group by definition of the quasi-inverse.

It is easy to verify that if \( f: \Gamma \to X \) is a quasi-isometry, the map \( \varphi : \Gamma \to \mathrm{QI}(X) \) defined by \( \phi(\gamma) = f\gamma\bar f \) (where \( \gamma \) is seen as acting on \( \Gamma \) by left-translations) is a group morphism. Such a morphism may be called a quasi-action of \( \Gamma \) on \( X \) (compare with the definition in Kapovich’s lectures).

In fact we can say a bit more about the quasi-action of \( \Gamma \) on \( X \) coming from the quasi-isometry \( f \).

  1. Its image is uniform in the obvious sense: there are \( (K, C) \) such that every element \( g \in \varphi(\Gamma) \) can be represented by a \( (K, C) \)-quasi-isometry. (Note: in addition, the representative of \( \varphi(\gamma\eta) \) is at bounded distance from the product of that for \( \varphi(\gamma) \) with that for \( \varphi(\eta) \).)
  2. The quasi-action is cobounded, that is there exists \( B \) and \( x_0 \in X \) such that for every \( x \in X \) there exists \( \gamma \in \Gamma \) with \( d(x_0, \varphi(\gamma) x) \le B \) (where by \( \varphi(\gamma) \) we mean a \( (K, C) \)-quasi-isometric representative). (Note: this needs the uniformity of the action to make sense.)

Strong quasi-isometric-rigidity

For any \( X \) there is a natural morphism \( \mathrm{Isom}(X) \to \mathrm{QI}(X) \). The space \( X \) is said to be strongly QI-rigid if this is an isomorphism. This implies almost immediately that if \( X \) is proper then \( X \) is QI-rigid in the sense above: from \( \varphi \) we obtain a proper action on \( X \), and the coboundedness of the quasi-action translates into cocompactness.

In general the morphism \( \mathrm{Isom}(X) \to \mathrm{QI}(X) \) is neither injective nor surjective. For example if \( X = \mathbb{R}^n \) the image of the translations in \( \mathrm{QI}(X) \) is trivial. And \( X = \mathbb{H}^2 \) has many more classes of quasi-isometries than those of isometries (any lift of a diffeomorphism between nonisometric compact hyperbolic surfaces gives an example of a quasi-isometry not at bounded distance from any isometry).

However, when \( \mathrm{Isom}(X) \subset \mathrm{QI}(X) \) it is sometimes possible to conjugate \( \varphi(\Gamma) \) into \( \mathrm{Isom}(X) \).

Quasi-isometric rigidity of hyperbolic space

The proof of quasi-isometric rigidity for the hyperbolic space \( \mathbb{H}^{n+1} \), \( n \ge 2 \) proceeds as follows :

  • The visual boundary \( \partial\mathbb{H}^{n+1} \) is naturally identified with the sphere \( \mathbb{S}^n \) with its natural conformal structure. Then quasi-isometries of \( \mathbb{H}^{n+1} \) extend to \( \mathbb{S}^n \) as quasi-conformal homeomorphisms. The isometries are exactly those maps which identify with conformal maps. This gives group isomorphisms:
    \[
    \mathrm{QI}(\mathbb{H}^{n+1}) = \mathrm{QC}(\mathbb{S}^n), \: \mathrm{Isom}(\mathbb{H}^{n+1}) = \mathrm{Conf}(\mathbb{S}^n).
    \]
  • Tukia’s theorem can then be restated as follows: if \( \varphi \) is a uniform cobounded quasi-action of a discrete group \( \Gamma \) on \( \mathbb{H}^{n+1} \) then the subgroup \( \partial\varphi(\Gamma) \subset \mathrm{QC}(\mathbb{S}^n) \) can be conjugated into \( \mathrm{Conf}(\mathbb{S}^n) \).

Quasi-isometric rigidity of solvable groups

Let \( S \) be a solvable Lie group with a left-invariant Riemannian metric. Then there is an isometric embedding of \( S \) into a product \( H_1 \times \cdots \times H_r \) where \( H_i \) are homogeneous negatively curved Riemannian manifolds. For example there is an embedding of the 3-dimensional group \( \mathrm{Sol} \) into \( \mathbb{H}^2 \times \mathbb{H}^2 \) (see Whyte’s lectures). A scheme to attempt to prove that \( S \) is QI-rigid then goes roughly as follows:

  • There are boundaries \( \partial_i S = \partial H_i \setminus \{ \infty \} \) (where \( \infty \) is a point on \( \partial H_i \) depending on the embedding).
  • Ideally this gives an isomorphism
    \[
    \mathrm{QI}(S) = \prod_{i=1}^r \mathrm{Bilip}(\partial_i S)
    \]
    (where \( \partial_i S \) has a natural metric, for example the euclidean metric if \( H_i = \mathbb{H}^n \) so that \( \partial_i S = \mathbb{R}^n \)). This is a technical step which uses the coarse differentiation of Eskin–Fisher–Whyte.
  • The isometry group \( \mathrm{Isom}(S) \) is contained in the product of the « similarity groups » \( \mathrm{Sim}(\partial_i S) \) (in the case where \( H_i = \mathbb{H}^n \) this is just the group of Euclidean similarities). One needs to generalise Tukia’s theorem to be able to conclude that the image \( \varphi(\Gamma) \) in \( QI(S) = \prod_i \mathrm{Bilip}(\partial_i S) \) of a discrete group \( \Gamma \) quasi-isometric to \( S \) can be conjugated into \( \prod_i \mathrm{Sim}(\partial_i S) \).
  • There is a last step needed to conclude.

Lecture II

quasi-conformal maps

Uniform subgroups of \( \mathrm{QC} \)

An homeomorphism \( f : \mathbb{S}^n \to \mathbb{S}^n \) is said to be \( K \)-quasiconformal if it satisfies the following. For \( x \in \mathbb{S}^n \) and \( \varepsilon > 0 \) let
\[
K_\varepsilon(f)(x) = \sup_{d(y, x) = \varepsilon = d(z, x)} \frac{d(f(x), f(y))}{d(f(x), f(z))}
\]
(this measures « how much \( f \) distorts circles around \( x \) »). Then for all \( x \):
\[
\limsup_{\varepsilon \to 0} (K_\varepsilon(f)(x)) \le K.
\]

The set of all quasi-conformal homeomorphisms is a group which was denites by \( \mathrm{QC}(\mathbb{S}^n) \) in the previous lecture. A subgroup \( G \subset \mathrm{QC}(\mathbb{S}^n) \) is said to be uniform if there is some \( K > 0 \) such that all of its elements are \( K \)-quasiconformal. The proof that the boundary extensions of quasi-isometries of \( \mathbb{H}^{n+1} \) yield quasi-conformal maps of \( \mathbb{S}^n \) (see Kapovich’s lectures) also shows that the boundary action of a uniform group of quasi-isometries yields a uniform subgroup of \( \mathrm{QC}(\mathbb{S}^n) \).

Tukia’s theorem

The statement of Tukia’s theorem that will be proven is the following.

Theorem: Let \( n \ge 2 \) and \( G \subset \mathrm{QC}(\mathbb{S}^n) \) be a uniform subgroup. Suppose in addition that the action of \( G \) on distinct triples \( (a, b, c) \in (\mathbb{S}^n)^3 \) is cocompact. Then there exists \( g \in \mathrm{QC}(\mathbb{S}^n) \) such that \( g G g^{-1} \subset \mathrm{Conf}(\mathbb{S}^n) \).

The hypotheses are satisfied if \( G \) is the boundary extension of a uniform cobounded group of quasi-isometries of \( \mathbb{H}^{n+1} \) (we saw this previously for the uniformity, and the cocompacity follows from the fact that there is a proper quasi-equivariant map from the space \( T(\mathbb{S}^n) \) of distinct triples to \( \mathbb{H}^{n+1} \), defined for example by associating to a triple the center of gravity of the ideal triangle having this triple as vertices).

Characterisation of conformal maps

The following characterisation of conformal maps among merely quasiconformal ones is very natural to state but nontrivial to prove. It follows for example from Gehring’s theorem stated in Kapovich’s lectures.

Theorem: A 1-quasiconformal map is conformal.

Radial points

Let \( G \subset \mathrm{QC}(\mathbb{S}^n) \). We view it as quasi-acting on \( \mathbb{H}^{n+1} \) and we fix a basepoint \( x_0 \in \mathbb{H}^{n+1} \). Let \( x \in \mathbb{S}^n \); it is called a radial point (or sometimes conical limit point) for \( G \) if there exists a geodesic ray \( \rho \) in \( \mathbb{H}^{n+1} \) such that \( \partial\rho = \{ x \} \), and a sequence \( g_i \in G \) such that \( g_i x_0 \) converges to \( x \) in \( \mathbb{H}^{n+1} \cup \partial \mathbb{H}^{n+1} \) and stays within bounded distance of \( \rho \).

If \( G \) acts cocompactly on \( T(\mathbb{S}^n) \) (equivalently its quasi-action on \( \mathbb{H}^{n+1} \) is cobounded) then every point in \( \mathbb{S}^n \) is a radial point.

Measurable conformal structures

Recall that an homeomorphism is quasiconformal if and only if it is differentiable at almost every point in \( \mathbb{S}^n \) and the ratio between the maximal eigenvalues of its derivatives is bounded. To take advantage of this we introduce a space of conformal structures on \( \mathbb{S}^n \).

We will consider objects which are defined only almost everywhere so it is more convenient to work on \( \mathbb{R}^n \) rather than \( \mathbb{S}^n = \mathbb{R}^n \cup \{\infty \} \). A measurable conformal structure on \( \mathbb{R}^n \) is a measurable map
\[
\mu : \mathbb{R}^n \to S := \{ n\times n\text{ symmetric positive definite matrices with determinant } 1 \}.
\]
Some remarks:

  • We may identify \( S \) with the Riemannian symmetric space \( \mathrm{SL}(n, \mathbb{R}) / \mathrm{SO}(n) \).
  • If \( \mu(x) = \mathrm{Id} \) for almost every \( x \) we say that \( \mu \) is the standard conformal structure.

An application \( f : \mathbb{R}^n \to \mathbb{R}^n \) which is differentible at almost every point is said to be conformal between two structures \( \mu \) and \( \mu’ \) if
\[
\mu(x) = Df_x^T \cdot \mu'(x) \cdot Df_x
\]
for almost every \( x \). For \( \mu = \mu’ \) the standard conformal structure in \( \mathbb{R}^n \) we get back the usual notion of conformal map.

For a matrix \( A \in S \) with largest eigenvalue \( \lambda_+ \) and smallest one \( \lambda_- \) let
\[
K(\mathrm{Id}, A) = \frac n 2 \max(\log(\lambda_+), -\log(\lambda_-))
\]
and extend it to a \( \mathrm{SL}(n, \mathbb{R}) \)-invariant function on \( S \times S \). Then a map is \( K \)-quasiconformal between \( \mu \) and \( \mu’ \) if and only if it is differentiable almost everywhere and
\[
\exp(K(\mu(x), Df_x^T\cdot \mu'(x) \cdot Df_x)) \le K
\]
for almost every \( x \). The notion of quasiconformality does not depend on the conformal structure. The first step in proving Tukia’s theorem will be to find a conformal structure \( \mu \) which is preserved by \( G \).

If \( f \) is quasiconformal let
\[
\mu_f = Df^T \cdot Df
\]
which is a conformal structure on \( \mathbb{R}^n \). We have the following equivariance property :
\[
\forall g \in \mathrm{QC}(\mathbb{R}^n) : \mu_{fg} = Dg^T \cdot (\mu_f \circ g) \cdot Dg.
\]
Now define :
\[
M_x = \{ \mu_f(x) : f \in G \} \subset S.
\]
Because \( G \) is uniform this is a bounded subset of \( S \) (with the distance given by \( K \)). In addition it is \( G \)-invariant under the action given above, and by the Cartan–Tits fixed point theorem it follows that \( G \) has a fixed point in \( S \) (the circumcenter of \( M_x \)). Denote by \( \mu_x \) this fixed point; then \( \mu : x \mapsto \mu_x \) is a \( G \)-invariant conformal structure.

It remains to prove that we can pass from \( G \) preserving \( \mu \) to a conjugate of \( G \) preserving the standard structure. In dimension 2 this follows from Sullivan’s theorem which says that any conformal structure is mapped by a quasiconformal map to the standard one. In higher dimension another argument is needed.

Lecture III

Recall that we have a uniform subgroup \( G \subset \mathrm{QC}(\mathbb{S}^n) \) preserving a measurable conformal structure \( \mu \). In addition we know that \( G \) extends to a cobounded quasi-action on \( \mathbb{H}^{n+1} \), and this implies that all points are radial.

We will make use of the following notion.

Definition: Let \( X, Y \) be topological spaces and \( f : X \to Y \) a Borel map. Let \( m \) be a Borel measure on \( X \). The map \( f \) is said to be \( m \)-approximately continuous at a point \( x \in X \) if the following holds:
\[
\forall \varepsilon > 0 : \lim_{r \to 0} \left( \frac{m(y \in B(x, r) : d_Y(f(x), f(y)) \ge \varepsilon)} {m(B(x, r))} \right) = 0.
\]

Any Borel map is in fact \( m \)-approximately continuous at \( m \)-almost every point. This applies to the map \( \mu : \mathbb{S}^n \to S \) (for the Lebesgue measure on \( \mathbb{S}^n \)), and so Tukia’s theorem can be reformulated as follows.

Theorem: Soit \( G \) a uniform subgroup of \( \mathrm{QC}(\mathbb{S}^n) \). Suppose that there exists a \( G \)-invariant measurable conformal structure \( \mu \) on \( \mathbb{S}^n \), and a point \( x \in \mathbb{S}^n \) at which \( \mu \) is approximately continuous, and which is radial for \( G \). Then there exists \( f \in \mathrm{QC}(\mathbb{S}^n) \) such that \( fGf^{-1} \subset \mathrm{Conf}(\mathbb{S}^n) \).

The proof of this statement is a « zooming » argument similar to what was used in the proof of Schwartz’s theorem in Kapovich’s lectures.

Somme lemmas

For \( A \in S \) let \( K(A) = e^{K(\mathrm{Id}, A)} \); for \( g \in \mathrm{QC}(\mathbb{S}^n) \) let \( K(g, x) = K( (Dg_x)^T \cdot Dg_x) \).

Lemma 1: Let \( K > 0 \).

  1. Let \( f_i \) be a sequence of \( K \)-quasiconformal applications. Suppose that \( f_i \to f \) pointwise and:
    \[
    \forall \varepsilon > 0 : \lim_{i \to +\infty} (\mathrm{Leb}(x : K(f_i, x) \ge 1 + \varepsilon)) = 0.
    \]
    Then \( f \) is conformal.
  2. Let \( (f_i)_{i \ge 0} \) be relatively compact in \( \mathrm{QC}(\mathbb{S}^n) \) (in the pointwise convergence topology), and let \( E_i \) a sequence of Borel sets with \( \mathrm{Leb}(E_i) \to 0 \). Then we have also \( \mathrm{Leb}(f_i(E_i))\to 0 \).

Lemma 2: Fix \( \lambda, c \) and let \( f_i \in \) be a sequence of \( (\lambda, c) \)-quasi-isometries, such that in addition there exists a point \( y \in \mathbb{H}^{n+1} \) such that \( f_i(y) \) is bounded. Then \( (f_i) \) is relatively compact.

Proof of Tukia’s theorem

Let \( \mu \) be a \( G \)-invariant conformal structure. We may assume that \( \mu \) is approximately continuous at 0 and that \( \mu(0) = \mathrm{Id} \). Let \( y \in \mathbb{H}^{n+1} \). We also know that 0 is radial for \( G \) and it follows that there exists a sequence \( g_i \in G \) such that \( g_iy \to 0 \) and \( g_i y \) stays within bounded distance of the geodesic \( [0, \infty] \) in \( \mathbb{H}^{n+1} \).

It follows that there exists a sequence \( T_i \in \mathrm{Isom}(\mathbb{H}^{n+1}) \) such that \( T_iz = t_i z \) for some \( t_i > 0 \) with \( t_i \to +\infty \) and \( T_i g_iy \) is bounded. By Lemma 2 the sequence \( f_i = T_ig_i \) has an accumulation point \( f \in \mathrm{QC}(\mathbb{S}^n) \). The rest of the proof is dedicated to show that
\[
fGf^{-1} \subset \mathrm{Conf}(\mathbb{S}^n)
\]
which concludes the proof of Tukia’s theorem.

To prove this claim we need to show that for any \( g \in G \) the sequence \( h_i = f_i g f_i^{-1} \) converges to a conformal map. We let
\[
\mu_i = (Df_i^{-1})^T \cdot \mu \cdot Df_i^{-1}
\]
then \( h_i \) is conformal for \( \mu_i \). On the other hand, since \( g_i \in G \) preserves \( \mu \) we have in fact that
\[
(\ast) \qquad \mu_i(x) = ((DT_i^{-1})_x)^T \cdot \mu(x) (DT_i^{-1})_x = \mu(x/t_i^2).
\]
Now it is easy to see that
\[
(\dagger) \qquad K(h_i, x) \le K(\mu_i(x)) K(\mu_i(g_ix)).
\]
The approximate continuity of \( \mu \) at 0 gives:
\[
\frac{\mathrm{Leb}(u \in B(0, r) : K(\mu(y)) \ge 1 + \varepsilon)}{r^n} \underset{r \to 0}{\rightarrow} 0
\]
and it follows for each \( i \) that:
\[
\frac{\mathrm{Leb}(u \in B(0, r) : K(\mu_i(y)) \ge 1 + \varepsilon)}{r^n} \underset{r \to 0}{\rightarrow} 0.
\]
Now using a diagonal argument together with \( (\ast) \) and the fact that \( t_i^{-2} \to 0 \) we obtain that, fixing \( R > 0 \) and putting
\[
A_i = \{ y \in B(0, R) : K(\mu_i(y)) \ge 1 + \varepsilon \}
\]
we have \( \mathrm{Leb}(A_i) \to 0. \). Now putting \( B_i = h_iA_i \) we have
\[
B_i = \{ y \in B(0, R) : K(\mu_i(g_i y)) \ge 1 + \varepsilon \}
\]
On the other hand, by Lemma 1.2 we have that \( \mathrm{Leb}(B_i) \to 0 \). By \( (\dagger) \) we have that
\[
\mathrm{Leb}(y \in B(0, R) : K(h_, y) \ge 1 + \varepsilon) \le \mathrm{Leb}(A_i) + \mathrm{Leb}(B_i)
\]
which implies that the left-hand side goes to 0. Applying Lemma 1.1 yields that \( h_i \) converges to a conformal map on \( B(0, R) \), and since \( R \) was arbitrary this concludes the proof.

Remarks

Other symmetric spaces

Tukia’s theorem can be extended to groups of quasiconformal maps on the boundary of complex hyperbolic space (this is due to Richard Chow, and has a proof similar to the one described above). The other rank 1 symmetric spaces are actually strongly quasi-isometrically rigid, that is \( \mathrm{QI}(X) = \mathrm{Isom}(X) \) (this is a theorem of Pansu).

Bi-Lipschitz homeomorphisms

The group \( \mathrm{Bilip}(\mathbb{R}^n) \) of bi-Lipschitz maps of Euclidean space is contained in \( \mathrm{QC}(\mathbb{S}^n) \). A subgroup \( G \subset \mathrm{Bilip}(\mathbb{R}^n) \) is said to be uniform if there exists a \( R > 0 \) such that:
\[
\forall g \in G : \exists K \ge k > 0 : K/k \le R \text{ and } \forall x, y \in \mathbb{R}^n, k|x – y| \le |gx – gy| \le K|x – y|.
\]
A corollary of Tukia’s theorem is then the following: if \( G \subset \mathrm{Bilip}(\mathbb{R}^n) \) is uniform and every point in \( \mathbb{R}^n \) is radial for \( G \) then there exists \( f \in \mathrm{Bilip}(\mathbb{R}^n) \) such that \( fGf^{-1} \subset \mathrm{Sim}(\mathbb{R}^n) \) (where \( \mathrm{Sim} \) is the group of similarities).

This follows from two simple observations: that \( \mathrm{Sim} = \mathrm{Bilip} \cap \mathrm{Conf} \), and that uniform subsets of \( \mathrm{Bilip} \) are closed in the topology of pointwise convergence on \( \mathrm{QC}(\mathbb{S}^n) \).

Results similar to the above for subgroups of homeomorphisms of the circle or the real line are due to Hinkkanen (in Uniformly quasisymmetric groups) and Farb–Mosher (in Quasi-isometric rigidity for the solvable Baumslag-Solitar groups. II).

Lecture IV

Let \( h \) be the function \( (x_1, \ldots, x_{n+1}) \in \mathbb{H}^{n+1} \mapsto x_n \) (« height function » on \( \mathbb{H}^{n+1} \)). Let :
\[
\mathrm{QI}_\mathrm{hr}(\mathbb{H}^{n+1}) = \{ f \in \mathrm{QI}(\mathbb{H}^{n+1}) : \exists C_f \in \mathbb{R}, h \circ f – h = C_f + O(1) \}.
\]
Then the boundary extension of a map in \( \mathrm{QI}_\mathrm{hr} \) is a bi-Lipschitz homeomorphism of \( \mathbb{R}^n \).

A subgroup \( G \subset \mathrm{QI}_\mathrm{hr} \) never acts coboundedly on \( \mathbb{H}^{n+1} \). Rather, the condition to consider on these groups is that their boundary action is cocompact on distinct pairs.

Solvable groups and negatively curved homogeneous spaces

There is a natural identification of the solvable group \( \mathbb{R}^n \times_{e^t} \mathbb{R} \) with the hyperbolic space \( \mathbb{H}^{n+1} \) where it acts simply transitively as a group of isometries fixing \( \infty \). In fact the natural left-invariant metric on this group is the hyperbolic metric.

Similarly, complex hyperbolic space \( \mathbb{C}\mathbb{H}^{n+1} \) is naturally isometric to the group \( N \times_{\phi^t} \mathbb{R} \) where \( N \) is the \( 2n + 1 \)-dimensional Heisenberg group and \( \phi \) is a specific automorphism. For \( n = 1 \) the group \( N \) can be realised as the group of matrices
\[
\left(\begin{array}{ccc} 1 & x & z \\ 0 & 1 & y \\ 0 & 0 & 1 \end{array} \right), \, x, y, z \in \mathbb{R}
\]
and \( \phi^t \) is given by
\[
\phi^t\left(\begin{array}{ccc} 1 & x & z \\ 0 & 1 & y \\ 0 & 0 & 1 \end{array} \right) = \left(\begin{array}{ccc} 1 & e^tx & e^{2t}z \\ 0 & 1 & e^ty \\ 0 & 0 & 1 \end{array} \right).
\]
The quaternionic hyperbolic spaces and octonionic hyperbolic plane can be realised in similar ways.

In general a theorem of Heintze states that a negatively curved homogeneous Riemannian manifold must be of the form \( N \times_{\phi^t} \mathbb{R} \) where \( N \) is a nilpotent group and \( \phi \) a « dilating » automorphism.

The simplest example is when \( N = \mathbb{R}^n \) is abelian. Then we must have \( \phi^t = e^{tA} \) where \( A \) is a matrix with all its eigenvalues in \( ]0, +\infty [ \). Let \( H_A \) be the space \( \mathbb{R}^n \times_{e^{tA}} \mathbb{R} \). The following examples are illustrative of the geometry of this space in different situations:

  • If \( A \) is scalar, \( A = \lambda\mathrm{Id} \) for somes \( \lambda > 0 \) then \( H_A = \mathbb{H}_c^{n+1} \) is the space of constant curvature \( c < 0 \).
  • If \( A \) is diagonal with diagonal terms \( \alpha_1, \ldots, \alpha_r > 0 \) with multiplicities \( m_1, \ldots, m_r \) then \( H_A \) contains totally geodesic copies of the spaces \( \mathbb{H}_{c_i}^{m_i} \).
  • If \( A \) has nontrivial Jordan blocks, for example \( A = \left(\begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array}\right) \), then the structure is more complicated.

On every space \( H = N \times \mathbb{R} \) there is a height function \( h \) obtained by projectin into the \( \mathbb{R} \) factor. The goal is to study the group \( \mathrm{QI}_\mathrm{hr}(H) \) of quasi-isometries respecting \( h \), which is defined as above, and the related boundary maps \( N \to N \).

Conjecture (Xie, Cornulier, Carrasco): Let \( H \) be a negatively curved Riemannian homogeneous manifold. If \( H \) is not isometric to a rank-one symmetric space then \( \mathrm{QI}(H) = \mathrm{QI}_\mathrm{hr}(H) \).

This conjecture is known to hold in particular when \( N \) is abelian.

The following conjecture relates the coarse geometry of solvable groups with that of products of homogeneous spaces. If \( H_i \) are homogeneous spaces with height functions \( h_i \) then an horocycle product in \( H_1 \times H_2 \) is the subset \( \{ (x_1, x_2) : h_1(x_1) + h_2(x_2) = 0 \} \).

Conjecture (Eskin–Fisher–Whyte): Let \( H_i \) be negatively curved homgeneous spaces. If \( G \subset H_1 \times H_2 \) is an horocycle product, then
\[
\mathrm{QI}(G) = \mathrm{QI}_\mathrm{hr}(H_1 \times H_2) \cap (\mathrm{QI}(H_1) \times \mathrm{QI}(H_2)).
\]

This is known in some cases by work of Eskin–Fisher–Whyte and Peng.

Boundary and quasiconformal maps

Let \( H = N \times_{\phi^t} \mathbb{R} \) be a negatively curved homogeneous space. The visual boundary \( \partial_\infty H \) has a distinguished point \( \infty \) and there is an identification \( \partial_\infty H = N \cup \{ \infty \} \). We will denote \( \partial H = N \).

Define a metric on \( \partial H \) as follows. Let \( x_1, x_2 \in \partial H \), then there is a unique \( t \in \mathbb{R} \) such that \( d_H((x_1, t), (x_2, t)) = 1 \). Put \( d(x_1, x_2) = e^t \).

On \( \mathbb{H}^{n+1} \) this gives the Euclidean metric on \( \mathbb{R}^n \) since \( d_H((x_1, t), (x_2, t)) = e^{-t}|x_1 – x_2| \). On \( \mathbb{H}_{-2}^{n+1} \) this yields the distance \( d(x_1, x_2) = |x_1 – x_2|^{\frac 1 2} \). In general it is harder (impossible?) to compute exactly. An example is \( H_A \) with \( A = \left( \begin{array}{cc} 2 & 0 \\ 0 & 1 \end{array} \right) \). There the distance on \( \mathbb{R}^2 \) is bi-Lipschitz to (but not equal to)
\[
d(x, y) = \max(|x_1 – y_1|, |x_2 – y_2|^{1/2}).
\]

Interlude

In view of QI-rigidity problem it is a natural question to ask when two homogeneous negatively curved spaces are quasi-isometric to each other. For spaces \( H_A, H_B \) this is the case of and only if, for every eigenvalue \( \lambda_i \) of \( A \), there is an eigenvalue \( \mu_i = e^{i\theta_i}\lambda_i \) of \( B \), and the Jordan blocks for \( A, \lambda_i \) and \( B, \mu_i \) are the same.

Bi-Lipschitz maps

In the case of the space \( H_A \) with \( A = \left(\begin{array}{cc} 1 & 0 \\ 0 & 2 \end{array} \right) \), any bi-Lipshitz map of \( H_A = \mathbb{R}^2 \times_{e^{tA}} \mathbb{R} \) decomposes as
\[
f(x, y) = (f_1(x, y), f_2(y)).
\]
The map \( f_2(y) \) must then be bi-Lipschitz. So must be the maps \( f_1(\cdot, y) \) for \( y \in \mathbb{R} \). On the other hand the maps \( f_1(x, \cdot) \) only have to be \( 1/2 \)-Hölder (see the distnce formula above for this space).

Suppose now that \( A \) is an \( n \times n \) diagonalisable matrix, with eigenvalues \( \alpha_1, \ldots, \alpha_r \) and corresponding multiplicities \( m_1, \ldots, m_r \). Let \( d_A \) be the distance defined above on \( \mathbb{R}^n = \partial H_A \). Let \( \pi \) be the map \( \mathbb{R}^n, d_A \to \mathbb{R}^{m_r}, d_{\alpha_r} \). Then, by an application of Tukia’s theorem, if \( G \subset \mathrm{Bilip}(\mathbb{R}^n, d_A) \) is uniform, the subbgroup \( \pi_*(G) \subset \mathrm{Bilip}(\mathbb{R}^{n_r}, d_r) \) is conjugated to a group of similarities.

Returning to the case of \( A = \left(\begin{array}{cc} 1 & 0 \\ 0 & 2 \end{array} \right) \) (where 1 and 2 may have multiplicities \( n_1, n_2 \)) we see that we can write a bi-Lipshitz map as
\[
f(x, y) = (f_1(x, y), \lambda_2 A_2(y + b_2) )
\]
where \( \lambda_2 > 0, b_2 \in \mathbb{R}^{n_2} \) and \( A_2 \in O(n_2) \).

A « foliated conformal structure » on \( H_A \) is a map \( \mu : \mathbb{R}^n \to S(n_1) \) where \( S(n_1) = \mathrm{SL}_{n_1}(\mathbb{R}) / \mathrm{SO}(n_1) \). It is possible to adapt Tukia’s arguments explained in the previous lecture to prove that in fact:
\[
f_1(x, y) = \lambda_1(y) A_1(y)(x + b_1(y))
\]
Further analysis shows that one can conjugate to the situation where \( A_1, \lambda_1 \) are constant. The final analogue of Tukia’s theorem that is obtained by these methods is the following statement (the similarity group of a metric space is just the group of maps which change distances by a constant factor).

Theorem (Dymarz–Xie): Let \( A \) be diagonalisable with positive eigenvalues and \( G \subset \mathrm{Bilip}(\mathbb{R}^n, d_A) \) be a uniform subgroup which acts cocompactly on distinct pairs. Suppose in addition that \( G \) is amenable, then there exists \( f \in \mathrm{Bilip}(\mathbb{R}^n, d_A) \) such that \( fGf^{-1} \subset \mathrm{Sim}(\mathbb{R}^n, d_A) \).

Application to QI-rigidity of solvable groups

Let \( \Gamma \) be an amenable discrete group. Suppose that there is a quasi-action \( \Gamma \to \mathrm{QI}(S) \). Then viewing \( S \subset H_1 \times H_2 \) the arguments above show that \( \Gamma \) can be conjugated into \( \mathrm{Sim}(\partial H_1 \times \partial H_2) \). There is still a last step to see that \( \Gamma \) is in fact contained in the isometry group.

Lecture V

Bi-Lipschitz and quasi-isometric equivalence

A quasi-isometry between two discrete groups is a bi-Lipschitz map if and only if it is bijective. The question of distinguishing between bi-Lipshitz and quasi-isometric equivalence thus boils down, in one direction, to finding a bijective map at finite distance from a given quasi-isometry.

Theorem (Whyte): Suppose \( G \) is a non-amenable group. Then any quasi-isometry \( G \to G’ \) from \( G \) to another discrete group \( G’ \) is at finite distance from a bijection.

On the other hand this is not true for amenable groups, for example a proper inclusion \( G’ \subset G \) with \( G \) amenable is never at finite distance from a bijection. On the other hand this does not precludes \( G \) and \’ G’ \) to be bi-Lipschitz to each other.

Exercise Prove this for the inclusion \( \mathbb{Z} \subset Z \times K \) where \( K \) is afinite group, and find a bi-Lipshitz map between the two.

Amenability

A group \( G \) is amenable if and only if there exists a Følner sequence in \( G \), that is a sequence \( F_i \) of finite subsets in \( G \) such that for any fixed finite subset \( K \subset G \) we have
\[
\lim_{i \to +\infty} \frac{|\partial_K S_i|}{|S_i|} = 0
\]
where
\[
\partial_K S_i = \{ g \in S_i : \exists k \in K, gk \not\in S_i \}.
\]
Følner sequences are an obstruction to finding a bijection close to an inclusion map: suppose \( S_i’, S_i \) are Følner sequence in \( G, G’ \) and that \( S_i’ \subset S_i \) has size \( \le (1-\delta)|S_i| \) (this is the situation for \( S_i’ \) coming from a subgroup of finite index) and \( i \) is large enough, there are not enough elements in \[ G’ \) which are close to \( S_i’ \) to fill the holes in \( S_i \) since \( |\partial_K S_i’| \le \varepsilon |S_i \setminus S_i’| \) where \( K \) can be taken arbitrarily large and \( \varepsilon \) arbitrarily small.

In general thre is the following criterion.

Whyte’s criterion: Let (\ X, Y \) be two finetely generated groups and \( \varphi : X \to Y \) a quasi-isometry. Then \( \varphi \) is at bounded distance from a bijection if and only if for any Følner sequence \( S_i \subset Y \) there exists \( r > 0 \) such that
\[
\sum_{x \in S_i} |\varphi^{-1}(\{x\}) – |S_i| = O(_partial_r S_i ).
\]
In general two maps \( \varphi, \psi \) are at bounded distance from each other if and only if
\[
\sum_{x \in S_i} (|\varphi^{-1}(\{x\}) – |\psi^{-1}(\{x\})|) = O(\partial_r S_i).
\]

Lamplighter groups

Recall that the wreath product \( G \wr \mathbb{Z} \) is the semi-direct product \( G^\mathbb{Z} \times \mathbb{Z} \) where a generator of \( \mathbb{Z} \) acts by \( (x_i) \mapsto (x_{i+1}) \).

Theorem (Dymarz): The groups
\[
G’ = (\mathbb{Z}/3 \times \mathbb{Z}/3) \wr \mathbb{Z}
\]
and
\[
G = \mathbb{Z}/3 \wr \mathbb{Z}
\]
are QI to each other (in fact there is an embedding \( G’ \subset G \) whise image has index 2) but there does not exist a bi-Lipschitz map \( G \to G’ \).

The Cayley graphs of \( G, G’ \) can be described as follows. Let \( T_d \) be the regular \( d \)-valent tree. Let \( DL(q, q) \) be the full subgraph of \( T_{q+1} \times T_{q+1} \) whose vertices are the pairs \( (x, y) \) such that \( h(x) + h(y) = 0 \) where \( h \) is a height function on \( T_{q+1} \). Then if \( G \) is a finite group of order \( q \) there is a natural generating set of \( G \wr \mathbb{Z} \) for which the Cayley graph is \( DL(q, q) \) (see e.g. this paper).

Thus, there is an edge between two vertices \( (x, y) \) and \( (x’, y’) \) if and only if \( h(x’) = h(x) + 1 \) and \( h(y’) = h(y) – 1 \) or the reverse.
(DESSIN!)

There is always a quasi-isometrically embedded copy of \( DL(q^2, q^2) \) inside \( DL(q, q) \) : fix a vertex in \( DL(q, q) \) and take all vertices at even distance from this one.
(DESSIN!)
In our case this embedding comes from an index 2 copy of \( G’ \) inside \( G \).

Boxes

A box in \( DL(q, q) \) of height \( L \) is a connected component of a « slice »:
\[
\{ (x, y) \in DL(q, q) : a \le h(x) \le a + L \}
\]
for some \( a \in \mathbb{Z}, L \in \mathbb N \). A quick computations shows that if \( S \) is a box of height \( L \) then \( |S| = (L+1) \cdot 3^L \) and \( |\partial_1 S = 2 \cdot 3^L \). Thus we see that a sequence of boxes with increasing height is a Følner sequence in \( DL(q, q) \).

\( DL(3, 3) \) and \( DL(9, 9) \) are not bi-Lipshitz to each other

Suppose that \( \bar\varphi \) is a bijective quasi-isometry from \( DL(9, 9) \) to \( DL(3, 3) \). We want to derive a contradiction.

Claim 1: \( \bar\varphi \) is at bounded distance from an application of the form \( i \circ \varphi \) where \( i \) is the inclusion \( DL(9, 9) \subset DL(3, 3) \) and \( \varphi \) is a 2-to-1 self-quasi-isometry of \( DL(9, 9) \).

Claim 2: There is no such \( \varphi \).

The goal in the rest of the lecture is to prove the second claim. This requires further study of quasi-isometries of \( DL(9, 9) \).

Structure of quasi-isometries

Let \( \mathbb{Q}_n \) be the set of ends of ends of \( T_{n+1} \), minus \( \infty \) (the end from which the height function \( h \) is taken) (in particular if \( n = p \) is prime \( \mathbb{Q}_n \) is identified with the \( p \)-adics integers, otherwise it is a priori just a commutative ring). There are « positive » and « negative » geodesic rays in \( DL(n, n) \) according to whether \( h \) is increasing or decreasing on them, and this yields an upper and lower boundaries which are both identified with \( \mathbb{Q}_n \).

Theorem (Eskin–Fisher–Whyte): Any quasi-isometry \( \varphi \) of \( DL(n, n) \) extends to a map on the boundary of the form
\[
\varphi(x, y) = ( \varphi_l(x), \varphi_u(y))
\]
where \( \varphi_l, \varphi_u \) are both bi-Lipshitz maps of \( \mathbb{Q}_n \).

On \( \mathbb{Q}_n \) there is a natural basis of clopen sets: a « ball » is given by the set of all points that can be reached from an inside point by a positive geodesic ray (if \( n = p \) these are the cosets of the additive subgroups \( p^m\mathbb{Z}_p \)). There is a natural measure \( \mu \) on the balls, where all balls associated to a point at height \( t \) have measure \( q^{-t} \). Similarly, the distance on \( \mathbb{Q}_n \) is defined by \( d(\xi, \eta) = q^{-t} \) if \(xi, \eta\) can be reached by geodesics rays originating from the same point at height \( t \).

Theorem (Cooper): If \( \varphi : \mathbb{Q}_n \to \mathbb{Q}_n \) is bi-Lipshitz then then there exists a ball \( B \) in \( \mathbb{Q}_n \) such that \( \varphi|_B \) is measure-linear, that is \( \varphi_*\mu|_B = \lambda\mu|_B \) for some \( \lambda > 0 \).

This is not enough to prove Claim 2, for this it will be needed that \( \varphi_u, \varphi_l \) be both measure-linear on the whole of \( \mathbb{Q}_n \). It is possible to get to this situation by using a « zooming » argument similar to the one in the last step of the proof of Tukia’s theorem. This yields \( \bar\varphi_u, \bar\varphi_l \) conjugated to the original maps, and which are measure-linear with coefficients \( \lambda_u, \lambda_l \).

Because of the structure of \( \mathbb{Q}_9 \) (compact-open sets are finite union of balls, and a ball is a disjoint union of 9 smaller balls) both \( \lambda_l \) and \( \lambda_u \) must be powers of 3.

Arguments above yield a quasi-isometry \( \psi \) of \( DL(9, 9) \) associated to \( \bar\varphi_l, \bar\varphi_u \) which is at finite distance from a conugate of \( \varphi \) and such that for any box \( S \) we have :
\[
\sum_{y \in S} |\psi^{-1}(\{y\})| = \lambda_l \lambda_u |S| + O(|\partial S|).
\]
On the other hand we have \( \sum_{y \in S} |\varphi^{-1}(\{y\})| = 2|S| \) and since \( \lambda_l \lambda_u \in 3^\mathbb{Z} \) this implies that
\[
\left| \sum_{y \in S} |\psi^{-1}(\{y\})| – \sum_{y \in S} |\varphi^{-1}(\{y\})| \right| \ge |S| + O(|\partial S|)
\]
which contradicts the fact that \( \psi \) and \( \varphi \) are at finite distance via Whyte’s criterion.

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