The asymptotic density of dead ends in non-amenable groups

TL;DR

This paper demonstrates that in non-amenable groups, the proportion of elements with large depth diminishes exponentially as depth increases, revealing a fundamental property of their structure.

Contribution

It establishes an exponential decay in the density of deep elements in non-amenable groups, providing new insights into their asymptotic properties.

Findings

Density of elements with depth ≥ d decreases exponentially with d

Quantitative characterization of element distribution in non-amenable groups

Supports understanding of geometric and algebraic properties of non-amenable groups

Abstract

We show that, in non-amenable groups, the density of elements of depth at least $d$ goes to $0$ exponentially in $d$.