A note on test elements for monomorphisms of free groups

TL;DR

This paper provides a geometric condition using Whitehead graphs and Cayley graph actions to identify test elements for monomorphisms in free groups.

Contribution

It introduces a new geometric criterion for constructing test elements in free groups, expanding understanding of their automorphism properties.

Findings

Provides a sufficient geometric condition for test elements

Uses Whitehead graph and Cayley graph actions in the construction

Enhances methods for analyzing monomorphisms in free groups

Abstract

A word in a group is called a test element if any endomorphism fixing it is necessarily an automorphism. In this note, we give a sufficient condition in geometry to construct test elements for monomorphisms of a free group, by using the Whitehead graph and the action of the free group on its Cayley graph.