Negligibility of parabolic elements in relatively hyperbolic groups

TL;DR

This paper proves that parabolic elements are negligible in density within finitely generated relatively hyperbolic groups, with the density decreasing exponentially fast, and extends this to commuting pairs, showing the group's degree of commutativity is zero.

Contribution

It establishes the zero density of parabolic elements and commuting pairs in relatively hyperbolic groups, providing exponential convergence rates and extending previous results.

Findings

Parabolic elements have zero density in the group.

The density of these elements converges exponentially fast.

The group's degree of commutativity is zero.

Abstract

We study density of parabolic elements in a finitely generated relatively hyperbolic group $G$ with respect to a word metric. We prove this density to be zero (apart from degenerate cases) and the limit defining the density to converge exponentially fast; this has recently been proven independently by W. Yang. As a corollary, we obtain the analogous result for the set of commuting pairs of elements in $G^2$, showing that the degree of commutativity of $G$ is equal to zero.