DRIFT OF RANDOM WALKS ON ABELIAN COVERS OF FINITE VOLUME HOMOGENEOUS SPACES

Let G be a connected simple real Lie group, Lambda(0) subset of G a lattice without torsion and Lambda (sic) Lambda(0) a normal subgroup such that Lambda(0)/Lambda similar or equal to Zd. We study the drift of a random walk on the Zd-cover Lambda \ G of the finite volume homogeneous space Lambda(0) \ G. This walk is defined by a Zariski-dense compactly supported probability measure mu on G. We first assume the covering map Lambda\G -> Lambda(0)\G does not unfold any cusp of Lambda(0)\G and compute the drift at 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.