Random walks on groups. Applications to Fuchsian groups (Q1078920)

Let G be a discrete finitely generated group. G is called transient if any symmetric, finitely supported and adapted probability measure on G is transient. Transience criteria are well known in the soluble case and the main result of this paper almost gives the answer in the general case and reads as follows: If there exists two finitely generated subgroups \(G\supset H_ 1\supset H_ 2\) such that \(| G:H_ 1| =| H_ 1:H_ 2| =| H_ 2| =+\infty\) then G is transient. Using former characterizations of a Fuchsian group of convergent type the author establishes relationships between the type of a subgroup \(\Gamma\) of a Fuchsian group \(\Gamma_ 0\) and the transience of \(\Gamma_ 0/\Gamma\).