Test elements in pro-$p$ groups with applications in discrete groups

TL;DR

This paper characterizes test elements in finitely generated profinite groups, especially free pro-$p$ groups, and explores their applications in discrete groups, providing explicit examples and establishing density results.

Contribution

It establishes a characterization of test elements in finitely generated profinite groups and applies this to free pro-$p$ groups and discrete groups, offering new explicit examples.

Findings

Test elements in finitely generated profinite groups are characterized by their absence from proper retracts.

Endomorphisms preserving automorphic orbits in free pro-$p$ groups are automorphisms.

The set of test elements in free discrete groups is dense in the profinite topology.

Abstract

Let $G$ be a group. An element $g \in G$ is called a test element of $G$ if for every endomorphism $\varphi:G \to G$, $\varphi(g)=g$ implies that $\varphi$ is an automorphism. We prove that for a finitely generated profinite group $G$, $g \in G$ is a test element of $G$ if and only if it is not contained in a proper retract of $G$. Using this result we prove that an endomorphism of a free pro-$p$ group of finite rank which preserves an automorphic orbit of a non-trivial element must be an automorphism. We give numerous explicit examples of test elements in free pro-$p$ groups and Demushkin groups. By relating test elements in finitely generated residually finite-$p$ Turner groups to test elements in their pro-$p$ completions, we provide new examples of test elements in free discrete groups and surface groups. Moreover, we prove that the set of test elements of a free discrete group of finite rank is dense in the profinite topology.