Typical elements in free groups are in different doubly-twisted conjugacy classes

TL;DR

This paper provides an algebraic criterion to distinguish elements in different doubly-twisted conjugacy classes in free groups and shows that this criterion holds with probability 1 for random choices.

Contribution

It introduces a simple algebraic condition for identifying distinct doubly-twisted conjugacy classes and proves its almost sure validity for random homomorphisms and elements.

Findings

Algebraic criterion for conjugacy class distinction

Criterion satisfied with probability 1 for random elements

Applicable to finitely generated free groups

Abstract

We give an easily checkable algebraic condition which implies that two elements of a finitely generated free group are members of distinct doubly-twisted conjugacy classes with respect to a pair of homomorphisms. We further show that this criterion is satisfied with probability 1 when the homomorphisms and elements are chosen at random.