On recent progress related to studying harmonic measures for random walks on hyperbolic isometry groups and lamplighter groups
Given a measured group $(G, \mu)$ acting on a nice space $X$, one might ask how to describe the
probability measures $\nu$ on $X$ which are invariant with respect to the action of $G$ on “average” with
respect to $\mu$. Turns out that such a measure ν usually is uniquely defined, depending on the choice
of $\mu$!
In my talk I will survey recent progress on understanding such measures with respect to actions of
isometry groups of hyperbolic spaces and lamplighter groups. In particular, we will discuss further
progress regarding obtaining the formulas for the Hausdorff dimensions of such measures. If time allows,
I will mention other research directions I have been pursuing recently. Joint
work with Eduardo Silva, Nikolay Bogachev and Giulio Tiozzo.