
## 2. Harmonic Measure

For a random walk on the fundamental group of a manifold of constant negative curvature, one has almost sure convergence to the boundary of hyperbolic space. This lets one define the hitting measure, by pushing forward the shift-space measure to the boundary S^n. How do properties of this measure relate to properties of the random walk?
1. ### Singularity of the Harmonic Measure

#### Problem 2.1.

[Vadim Kaimanovich] Let $\Gamma$ be a Fuchsian group, and let $\mu$ be a finitely supported measure on $\Gamma$. Is the harmonic measure singular with respect to the Lebesgue measure on the circle?
• ### Dimension of Harmonic Measure over Moduli Space

[Pablo Lessa] Fix some measure $\mu$ on $\pi_1(M)$, where $M$ is a manifold of curvature -1. Construct the random walk on hyperbolic $n$-space by picking a representation $\rho: \pi_1(M) \to$\text{Aut}(\mathbb{H}^n)$. How does the dimension of the harmonic measure$\nu$change as a function of$\rho$? Is the function$\rho \to -\log{\text{dim}(\nu_\rho)}$continuous? proper? • ### Witnessing a Conformal Measure #### Problem 2.3. [Francois Ledrappier] Let$G$be a$\delta$-hyperbolic group. Let$\nu$be a conformal measure on the Gromov Boundary$\partial_{\text{Gromov}} G$. does there exist some probability measure$\mu$on$G$such that$\nu$and the hitting measure for$\mu\$ lie in the same measure class?

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