Eduardo Reyes - Green metrics and metric structures on hyperbolic groups

Abstract

Teichmüller space is a classical construction that, for a given closed hyperbolic surface, parameterizes the geometric actions of its fundamental group on the hyperbolic plane. I will talk about a generalization of this space, where for an arbitrary Hyperbolic group we consider a metric space that parameterizes its geometric actions on Gromov hyperbolic spaces. Even in the surface group case, this space turns out to be much larger than Teichmüller space, and we can find points induced by negatively curved Riemannian metrics, Anosov representations, Random walks, geometric cubulations, etc. In particular, I will discuss how Green metrics (those encoding admissible random walks on the group) are dense in this space. This is joint work with Stephen Cantrell and Dídac Martínez-Granado.

Talk

Goal: $\Gamma$ finitely generated. We want to “understand the space of reasonable isometric actions of $\Gamma$“.

I Metric structures

We will have the following assumptions:

1. All actions are geometric, i.e. isometric, properly discontinuous, cobounded (quotient is bounded) Classical examples: Cayley-Graph, Groups acts as Fundamental group on compact space.
2. The group is hyperbolic and non-elementary Classical examples: Fundamental group of closed neg. curved. mfd.

Definition (Furman '02) $\Gamma$ be a Nonelementary hyperbolic group hyperbolic group. The space of metric structures $D_{\Gamma }$ is the space of all geometric actions $\Gamma \circlearrowright X$ on geodesic spaces up to an equivalence. Two spaces are equivalent if there is a $\Gamma$-equivariant $(A,\lambda )$-quasiisometry $f$ between $X$ and $Y$.

Let

This is supposed to be a generalisation of the Teichmüller-space. (nice!)

Definition

We define a Distance $\Delta ([X],[Y])$. This is determined by the $\lambda ,A$ of the maps above.

Example

1. Cayley graphs are elements in $D_{\Gamma }$
2. $\Gamma ={\pi }_1(\Sigma )$. Then $\Gamma$ acts on the Hyperbolic plane. It follows that the Teichmüller space is a subspace of $D_{\Gamma }$.
3. $\Gamma$ is the free group. Then the action of $\Gamma$ on a Baum shows that the Äußerer Automorphismus are a subspace of $D_{\Gamma }$.

Example

If $\Gamma$ admits a Projective pseudo-Anosov representation $p:\Gamma \to PS{L}_d(\mathbb{R})$ then $p$ also induces a point in $D_{\Gamma }$.

• openquestion How do different structures interact inside $D_{\Gamma }$?

Lemma

Cayley graphs are dense in $D_{\Gamma }$. It follows that $D_{\Gamma }$ is Separable space.

• openquestion Let $\Gamma$ be a Hyperbolic group and acting on a Cube complex. Are cubulations dense in the $D_{\Gamma }$ (Futer-Wise)
• openquestion Let $\Gamma$ be a Hyperbolic group and acting on a Cube complex. Are Projective pseudo-Anosov representations dense in $D_{\Gamma }$? (Doubz-Flechelle-Weissimm-Zhu ‘23)

II Green metrics

…

Green metrics are objects that interact with Random walk.

Theorem

Let $\Gamma$ be hyperbolic and not virtually free. There is no interaction between Cayley graphs and Green metrics.

• openquestion $\Gamma$ is a surface group. Then Teichmüller-space and Green metrics don’t talk to another, i.e. $\{[d_{\lambda }]|\lambda \text{ adm}\}\cap \text{Teich }\Gamma =\varnothing$ in $D_{\Gamma }$

The conjecture is that Teichmüller spaces are very special among all other spaces because they “cant be seen”.