## 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:

- 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?
- Is it known how large is the intersection of these two classes of metrics (considered up to rough similarity)?