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 $\Lambda ⊴{\Lambda }_0$ 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 $\Lambda \backslash G\to {\Lambda }_0\backslash G$ 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.

arXiv comment28 pages, 3 figures