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Some background (see e.g. the books by Ghys & de la Harpe or Bridson & Haefliger for more information):

Let $\Gamma$ be a group with a finite symmetric generating set $S$. Recall that $\Gamma$ is called a (word) hyperbolic group if the Cayley graph of $(\Gamma, S)$ is hyperbolic (in the sense of Gromov). This notion is independent of the choice of the generating set $S$.

Assuming $\Gamma$ to be hyperbolic, there is a finite directed graph canonically associated to $(\Gamma,S)$ called the geodesic automaton (GA), whose arrows are labelled by elements of $S$. Paths in the GA graph starting from a special basepoint correspond to segments of geodesics in the Cayley graph.

Now consider the recurrent vertices of the GA graph, i.e., the set of vertices that belong to loops. Following this paper by Haissinsky, Mathieu, and Mueller, let us say that the GA graph is strongly connected if given any two recurrent vertices, there is one path from one to the other.

For example, if $\Gamma = \mathbb{Z}$ and $S = \{1,-1\}$ then the GA graph is: GA graph of Z which is obviously not strongly connected.

Question. If $\Gamma$ is a nonelementary hyperbolic group, is the geodesic automaton graph strongly connected for some (all?) choice of $S$?

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If I understand correctly, you are asking if the automaton graph is recurrent, that is there is only one non trivial recurrence class (a recurrence class is trivial if it is reduced to one vertex).

For Fuchsian groups, you can always find a generating set such that the automaton graph is recurrent. This was proved by Caroline Series in the paper The infinite word problem and limit sets in Fuchsian groups (Ergodic theory and dynamical system, 1981).

However, in general, we do not know how to construct such an $S$. In general, applying Cannon's method, your graphs is not strongly connected for any $S$. There are counter example (I'll try to find some and add them as comments). However, some experts in the field think it is possible to find an $S$.

I strongly recommend the survey of Danny Calegari The ergodic theory of hyperbolic groups (available on his webpage).

If you want to do dynamics on the path space of the automaton, say you have a function defined on this path space, you can look at the so-called maximal components. Those are the recurrent components of you graph such that the spectral data of your function (usually the dominant eigenvalue of the associated transfer operator) is maximal. What can go wrong is if you have a path from one maximal component to another one (but not the other way around, otherwise these would not be two different components).

This was explored in the paper of Danny Calegari and Koji Fujiwara Combable functions,quasimorphisms, and the central limit theorem (Ergodic theory and dynamical systems, 2010). See also this paper of Sébastien Gouëzel Local limit theorem for symmetric random walks in Gromov-hyperbolic groups (Journal of the AMS, 2014).

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  • $\begingroup$ Thanks for the useful information. If I understood correctly, the question is open. $\endgroup$ – Jairo Bochi Sep 11 '18 at 12:30
  • $\begingroup$ I've been investigating a bit. The question is still open indeed, but experts think that you can always find a "good" $S$. I edited the answer accordingly. $\endgroup$ – M. Dus Sep 13 '18 at 14:11

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