Published on *Max Planck Institute for Mathematics* (http://www.mpim-bonn.mpg.de)

Posted in

- Talk [1]

Speaker:

Panagiotis Polymerakis
Affiliation:

MPIM
Date:

Thu, 14/11/2019 - 13:00 - 14:00 For a normal Riemannian covering $p \colon M \to N$ of a closed manifold $N$, we are interested in relations between the deck transformation group $\Gamma$ and the validity of the Liouville and the strong Liouville property on $M$. Our approach is probabilistic and relies heavily on the Brownian motion. Lyons and Sullivan constructed a discretization of the Brownian motion on $M$, obtaining a random walk on $\Gamma$. Their method was modified and extended in a work of Ballmann and Ledrappier, and in a recent joint work with Ballmann. In particular, it follows that there exists a probability measure $\mu$ on $\Gamma$ such that $H_{b}(M) \cong H_{b}(\Gamma, \mu)$ and $H^{+}(M) \cong H^{+}(\Gamma,\mu)$, where $H_{b}$ is the space of bounded harmonic functions and $H^{+}$ is the cone of positive harmonic functions. In this talk, we will survey some results that follow from this discretization. Finally, we will show that if $\Gamma$ is of exponential growth, then $M$ does not have the strong Liouville property, which was conjectured by Lyons and Sullivan.

**Links:**

[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39

[2] http://www.mpim-bonn.mpg.de/node/3444

[3] http://www.mpim-bonn.mpg.de/node/3050