Theorem. Any $\omega$-nilpotent cover of a recurrent Riemannian manifold is Liouville.
$\omega$-nilpotent ($\Gamma=\bigcup_{i=1}^{\infty}Z_{i}$, $Z_{i}$ normal in $\Gamma$, where $Z_{n+1}$ maps to the center of $\Gamma/Z_{n}$.)
The above theorem comes from a paper of T. Lyons and D. Sullivan. function theory random paths and covering spacesre (Theorem 2).
How to understand the $\omega$-nilpotent cover?
The second derived subgroup of the fundamental group of an open Riemann surface $X$ determines a covering space, is it $\omega$-nilpotent? (i.e. let $G=\pi_{1}(X)$, $G'=[G,G]$, $G''=[G',G']$, is the group $G/G''$ $\omega$-nilpotent?)
