
## 1. Boundaries

Relations between Poisson boundaries, Martin boundaries, Gromov Boundaries, Sublinearly Morse boundaries, etc.
1. ### Stability of the Liouville property

#### Conjecture 1.1.

For any group $G$ and two measures $\mu_1$, $\mu_2$ which are symmetric and finitely supported, the Poisson boundary for $(G, \mu_1)$ is trivial if and only if the Poisson boundary for $(G, \mu_2)$ is trivial.
• ### Poisson boundary for random walks on free (semi)-groups with infinite moment

#### Conjecture 1.2.

For any random walk on the free group or free semigroup, the Gromov boundary with the hitting measure is a topological model for the Poisson boundary.
1. Remark. [org.aimpl.user:kunal.chawla@mail.utoronto.ca] In the case that the measure $\mu$ has finite entropy and finite log-moment, this is a well-known result of Kaimanovich. This condition was weakened by Forghani and Tiozzo in the paper "Random Walks of Infinite Moment on Free semigroups" where they showed that for the free semigroup, it suffices to have either finite log-moment or finite entropy.

It would be interesting to extend this result to the free group, or to weaken it to a finite entropy condition.
• ### The Sublinearly Morse boundary and the Martin boundary

The Sublinearly Morse boundary of a geodesic metric space encodes the set of ’nearly-hyperbolic directions’. The Martin boundary for a measured group $(G, \mu)$ is the set of harmonic functions which appear as limits of ratios of Greens functions.

For certain ’nice’ random walks on hyperbolic groups one can identify the Gromov and Martin boundaries.

#### Problem 1.3.

[Ilya Gekhtman] If $G$ is a Hierarchically Hyperbolic Space, and $\mu$ is a symmetric non-degenerate probability measure on $G$ with finite support, then the Sublinearly Morse Boundary of $G$ embeds into the Martin boundary of $(G, \mu)$.

Cite this as: AimPL: Random walks beyond hyperbolic groups, available at http://aimpl.org/randwalkgroup.