The equation w(x,y)=u over free groups

TL;DR

This paper characterizes solutions to certain equations over free groups using advanced group theory, revealing complexities in solution structures and automorphism levels.

Contribution

It extends the understanding of equations in free groups by describing solutions to w(x,y)=u and highlighting cases requiring multiple automorphism levels.

Findings

Solutions to w(x,y)=u are characterized using existing group theory frameworks.

Some equations necessitate multiple automorphism levels for their solution description.

The paper provides an example of an equation with inherently complex solution structure.

Abstract

Using the theory developed by Olga Kharlampovich, Alexei Miasnikov, and, independently, by Zlil Sela to describe the set of homomorphisms of a f.g. group G into a free group F, we describe the solutions to equations with coefficients from F and unknowns x,y of the form w(x,y) = u, where u lies in F and w(x,y) is a word in {x,y}^\{pm 1}. We also give an example of a single equation whose solutions cannot be described with only one "level" of automorphisms.