The rank gradient and the lamplighter group

Allums, Derek; Grigorchuk, Rostislav

doi:10.2140/involve.2011.4.297

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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

DOI: 10.2140/involve.2011.4.297

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 $ℤ_{2} ≀ ℤ$ 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

Milestones

Received: 10 June 2011

Accepted: 11 June 2011

Published: 13 March 2012

Communicated by David R. Larson

Authors

Derek J. Allums
Texas A&M University College Station, TX 77843 United States

Rostislav I. Grigorchuk
Texas A&M University College Station, TX 77843 United States

Vol. 4, No. 3, 2011