Expected Depth of Random Walks on Groups

TL;DR

This paper investigates the expected depth of elements in a random walk on finitely generated groups, establishing criteria for convergence to a natural limit and exploring specific classes like nilpotent and Kazhdan Property (T) groups.

Contribution

It introduces criteria for the convergence of the expected depth of random walks on groups and characterizes this limit for certain classes of groups, including nilpotent and linear groups with Property (T).

Findings

Expected depth converges to a specific sum involving normal subgroups.

Convergence holds for nilpotent and Kazhdan Property (T) groups.

Provides examples where convergence does not occur.

Abstract

For $G$ a finitely generated group and $g \in G$, we say $g$ is detected by a normal subgroup $N \lhd G$ if $g \notin N$. The depth $D_G(g)$ of $g$ is the lowest index of a normal, finite index subgroup $N$ that detects $g$. In this paper we study the expected depth, $\mathbb E[D_G(X_n)]$, where $X_n$ is a random walk on $G$. We give several criteria that imply that $$\mathbb E[D_G(X_n)] \xrightarrow[n\to \infty]{} 2 + \sum_{k \geq 2}\frac{1}{[G:\Lambda_k]}\, ,$$ 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 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.