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

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.