Vol. 298, No. 2, 2019

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Expected depth of random walks on groups

Khalid Bou-Rabee, Ioan Manolescu and Aglaia Myropolska

Vol. 298 (2019), No. 2, 267–284
Abstract

For G a finitely generated group and g G, we say g is detected by a normal subgroup N G if gN. The depth DG(g) of g is the lowest index of a normal, finite index subgroup N that detects g. In this paper we study the expected depth, E[DG(Xn)], where Xn is a random walk on G. We give several criteria that imply that

E[DG(Xn)] n 2 + k2 1 [G : Λk],

where Λk is the intersection of all normal subgroups of index at most k. In particular, the equality holds in the class of all nilpotent groups and in the class of all linear groups satisfying Kazhdan’s property (T). We explain how the right-hand side above appears as a natural limit and also give an example where the convergence does not hold.

Keywords
expectation, residual finiteness growth, intersection growth, residual averages, depth function
Mathematical Subject Classification 2010
Primary: 05C81, 20F65, 60B15
Milestones
Received: 8 November 2017
Accepted: 10 July 2018
Published: 8 March 2019
Authors
Khalid Bou-Rabee
School of Mathematics
The City College of New York
New York, NY
United States
Ioan Manolescu
Departement de mathematiques
University of Fribourg
Fribourg
Switzerland
Aglaia Myropolska
Laboratoire de Mathématiques
Université Paris-Sud
Orsay
France