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