Solutions d'une équation quadratique dans le groupe libre. (Solutions of a quadratic equation in the free group) (Q1080517)

The author studies the set of solutions of a quadratic equation in a free group, i.e. of an equation of the form \(W=1\), where W is a quadratic word in the free generators \(x_ 1,x_ 2,...,x_ n\) of the free group F. By a solution it is meant, here, a couple \(<f,G>\) consisting of another free group G and a homomorphism \(f: F\to G\) such that \(Wf=1\). The main Theorem describes the set of solutions of such an equation, namely: A solution \(<f,G>\) is such that f is a homomorphism \(f: F\to G\) of the form \(f=\alpha sh\), where \(\alpha\) is an automorphism of F mapping W onto W or \(W^{-1}\), s a ''literal'' solution, i.e. a solution which maps \(X=\{x_ 1,x_ 2,...,x_ n\}\) into \(X^{\pm 1}\cup \{1\}=\{x_ 1,x_ 2,...,x_ n,x_ 1^{-1},x_ 2^{-1},...,x_ n^{-1},1\}\) and h a homomorphism of F onto G. The method of proof is combinatorial.