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

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.

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Mathematical Subject Classification 2010

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