The rank gradient and the lamplighter group

Abstract

We introduce the notion of the rank gradient function of a descending chain of subgroups of finite index and show that the lamplighter group ${\mathbb{Z}}_2\wr \mathbb{Z}$ has uncountably many 2-chains (that is, chains in which each subsequent group has index 2 in the previous group) with pairwise different rank gradient functions. In doing so, we obtain some information on subgroups of finite index in the lamplighter group.