#### Vol. 4, No. 3, 2011

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The rank gradient and the lamplighter group

### Derek J. Allums and Rostislav I. Grigorchuk

Vol. 4 (2011), No. 3, 297–305
##### 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 ${ℤ}_{2}\wr ℤ$ 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.

##### Keywords
lamplighter group, rank gradient, decay of rank gradient, finitely generated residually finite amenable groups
##### Mathematical Subject Classification 2010
Primary: 20E18, 20E22, 20E26, 20F65