#### 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 $g\in G$, 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

$\mathbb{E}\left[{D}_{G}\left({X}_{n}\right)\right]\underset{n\to \infty }{\overset{}{\to }}2+\sum _{k\ge 2}\frac{1}{\left[G:{\Lambda }_{k}\right]}\phantom{\rule{0.3em}{0ex}},$

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