Drift of random walks on abelian covers of finite volume homogeneous spaces

Abstract

Let $G$ be a connected simple real Lie group, $\Lambda_{0}\subseteq G$ a lattice and $\Lambda \unlhd \Lambda_{0}$ a normal subgroup such that $\Lambda_{0}/\Lambda\simeq \mathbb{Z}^d$. We study the drift of a random walk on the $\mathbb{Z}^d$-cover $\Lambda\backslash G$ of the finite volume homogeneous space $\Lambda_{0}\backslash G$. This walk is defined by a Zariski-dense compactly supported probability measure $\mu$ on $G$. We first assume the covering map $\Lambda\backslash G\rightarrow \Lambda_{0}\backslash G$ does not unfold any cusp of $\Lambda_{0}\backslash G$ and compute the drift at \emph{every} starting point. Then we remove this assumption and describe the drift almost everywhere. The case of hyperbolic manifolds of dimension 2 stands out with non-converging type behaviors. The recurrence of the trajectories is also characterized in this context.