How much do the classes of geodesic metrics and Green metrics on a Gromov hyperbolic group differ?

Question

Background

Let $G$ be a Gromov hyperbolic group. If $G$ acts properly discontinuously and cocompactly on a proper geodesic metric space $X$, then any bijective identification of $G$ with its orbit induces a metric on $X$ (changing the orbit leads to a bounded perturbation of the metric). I will call such metrics on $G$ geodesic. This class includes for instance the word metrics associated to finite generating sets.

On the other hand, let $\mu$ be a symmetric probability measure on $G$. Denote by $F(x,y)$ the probability that the corresponding random walk starting at $x$ will ever hit $y$. Following Blachere, Haissinsky, & Mathieu, we assume two technical conditions, satisfied by all finitely supported measures:

• $\mu$ has exponential moment, i.e. $\sum_{g\in G} e^{\lambda \lvert{g}\rvert} \mu(g) < \infty$ for some $\lambda > 0$ (the length of $g$ is considered with respect to any word metric on $G$)
• for any $r>0$ there exists a constant $C(r)$ such that $F(x,y) \leq C(r) F(x,v)F(v,y)$ whenever $v$ is within distance $r$ from a geodesic segment joining $x$ and $y$ in a fixed Cayley graph of $G$

Under these assumptions the Green metric $d_G(x,y) = -\log F(x,y)$ is hyperbolic, left-invariant and quasi-isometric to the word metrics through the identity map. (This is the metric on $G$ with respect to which the hitting probability of the random walk is a quasi-conformal measure on the boundary).

Finally, we say that two metrics $d_1, d_2$ on $G$ are roughly similar if there exists $\alpha > 0$ such that the function $\lvert d_1-\alpha d_2 \rvert$ is bounded. This is an appropriate equivalence relation if we are interested in the measurable structure on the boundary, as roughly similar metrics give rise to Hölder equivalent visual metrics and the same class of Patterson-Sullivan measures.

Questions

Above, I described two large classes of metrics on a Gromov hyperbolic group. My question is: how different are they in terms of rough similarity? (Blachere, Haissinsky, & Mathieu say that the Green metrics are usually not geodesic, but they don't provide more details on this matter). More precisely:

1. Are there any nice examples of metrics in one of these classes, for which it is easy to see that they are not in the other class?
2. Is it known how large is the intersection of these two classes of metrics (considered up to rough similarity)?

Answer 1

In general, for a finitely supported measure $\mu$ whose support generates $\Gamma$, the Green metric and the word metric are not equivalent (in the sense you give, that is they are not roughly similar). Actually, you only need superexponential moments and not finite support. More precisely, if the two metrics are roughly similar, then $\Gamma$ is virtually free. This is Theorem 1.5 in the paper of Gouëzel, Matheus and Maucourant entropy and drit in word hyperbolic groups (Inventiones, 2018).

A rough argument is as follows. Equivalence means that $|d_1-ad_2|\leq C$ for some $a$ and some $C$. Suppose that $C=0$. Then, you have $d_1=ad_2$ and so the horofunction boundaries for both metric must coincide. However, the horofunction boundary for the word metric must be totally disconnected, whereas the horofunction for the Green metric (the so-called Martin boundary) is not, since it coincides with the Gromov boundary (except for the particular case where $\Gamma$ is virtually free). You can make this discussion precise, even when $C\neq 0$, using some asymptotic version of the above arguments. One way to do so is to look at the Martin cocyle and to compare it with the Buseman cocyle for the word distance.

For free groups, it can happen that the two metrics are roughly similar. As you say, they are roughly similar if and only if the corresponding Patterson-Sullivan measures are equivalent, if and only if the Hausdorff dimension of the hitting measure on the Gromov boundary is maximal (this is contained in the paper by Blachere Haïssinsky and Mathieu you refer to). So basically, you have to compare the Hausdorff dimension of the Gromov boundary and the Hausdorff dimension of the hitting measure for the random walk.

For probability measures supported on the standard set of generators on a free group, computations can be made and the answer is as follows: the hitting measure has maximal dimension if and only if the random walk is the simple random walk (each generator has the same weigth). This is described with many details in Ledrappier's survey : Some asymptotic properties of random walks on free groups.

However, for general probability measures with an exponential moment, this is still widely open to know if the two metrics are roughly similar, even in free groups. Also, I don't think that we even know the answer for (non-nearest neighbor) finitely supported random walks on free groups.