Expected depth of random walks on groups

Khalid Bou-Rabee, Ioan Manolescu and Aglaia Myropolska

Vol. 298 (2019), No. 2, 267–284

DOI: 10.2140/pjm.2019.298.267

Abstract

For $G$ a finitely generated group and $g\in G$, we say $g$ is detected by a normal subgroup $N◃G$ if $g\notin N$. The depth $D_G\left(g\right)$ of $g$ is the lowest index of a normal, finite index subgroup $N$ that detects $g$. In this paper we study the expected depth, $\mathbb{E}\left[D_G\left(X_n\right)\right]$, where $X_n$ is a random walk on $G$. We give several criteria that imply that

$$\mathbb{E}\left[D_G\left(X_n\right)\right]\underset{n\to \infty }{\overset{}{\to }}2+\sum _{k\ge 2}\frac{1}{\left[G:{\Lambda }_k\right]},$$

where ${\Lambda }_k$ 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 $\left(T\right)$. 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