Asymptotic density of test elements in free groups and surface groups

TL;DR

This paper investigates the distribution of test elements in free and surface groups, proving they form a dense and positively dense subset, thus answering a longstanding question in group theory.

Contribution

It establishes that the set of test elements is a net with positive asymptotic density and dense in the profinite topology for certain groups, advancing understanding of their algebraic structure.

Findings

Test elements form a net in free and surface groups.

The set of test elements has positive asymptotic density.

Test elements are dense in the profinite topology.

Abstract

An element $g$ of a group $G$ is a test element if every endomorphism of $G$ that fixes $g$ is an automorphism. Let $G$ be a free group of finite rank, an orientable surface group of genus $n \geq 2$, or a non-orientable surface group of genus $n \geq 3$. Let $\mathcal{T}$ be the set of test elements of $G$. We prove that $\mathcal{T}$ is a net. From this result we derive that $\mathcal{T}$ has positive asymptotic density in $G$. This answers a question of Kapovich, Rivin, Schupp, and Shpilrain. Furthermore, we prove that $\mathcal{T}$ is dense in the profinite topology on $G$.