Remnant inequalities and doubly-twisted conjugacy in free groups

TL;DR

This paper introduces new criteria and algorithms for determining doubly-twisted conjugacy in free groups, showing that such conjugacy is rare under certain conditions and analyzing the density of homomorphisms.

Contribution

It provides a remnant inequality condition for non-conjugacy, an algorithm for deciding conjugacy, and analyzes the density of homomorphisms in free groups.

Findings

Probability that two words are not doubly-twisted conjugate is 1 under certain conditions.

Algorithm can decide conjugacy relations when remnant conditions are met.

Computed densities and expected values of homomorphisms in free groups.

Abstract

We give two results for computing doubly-twisted conjugacy relations in free groups with respect to homomorphisms $\phi$ and $\psi$ such that certain remnant words from $\phi$ are longer than the images of generators under $\psi$. Our first result is a remnant inequality condition which implies that two words $u$ and $v$ are not doubly-twisted conjugate. Further we show that if $\psi$ is given and $\phi$, $u$, and $v$ are chosen at random, then the probability that $u$ and $v$ are not doubly-twisted conjugate is 1. In the particular case of singly-twisted conjugacy, this means that if $\phi$, $u$, and $v$ are chosen at random, then $u$ and $v$ are not in the same singly-twisted conjugacy class with probability 1. Our second result generalizes Kim's "bounded solution length". We give an algorithm for deciding doubly-twisted conjugacy relations in the case where $\phi$ and $\psi$ satisfy a similar remnant inequality. In the particular case of singly-twisted conjugacy, our algorithm suffices to decide any twisted conjugacy relation if $\phi$ has remnant words of length at least 2. As a consequence of our generic properties we give an elementary proof of a recent result of Martino, Turner, and Ventura, that computes the densities of injective and surjective homomorphisms from one free group to another. We further compute the expected value of the density of the image of a homomorphism.