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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


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      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


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          Problem 2.2.

          [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


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              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.