Some quadratic equations in the free group of rank 2

For a given quadratic equation with any number of unknowns in any free group F , with right-hand side an arbitrary element of F , an algorithm for solving the problem of the existence of a solution was given by Culler [8] using a surface method and generalizing a result of Wicks [46]. Based on different techniques, the problem has been studied by the authors [11; 12] for parametric families of quadratic equations arising from continuous maps between closed surfaces, with certain conjugation factors as the parameters running through the group F . In particular, for a oneparameter family of quadratic equations in the free group F2 of rank 2, corresponding to maps of absolute degree 2 between closed surfaces of Euler characteristic 0, the problem of the existence of faithful solutions has been solved in terms of the value of the self-intersection index W F2 ! ZŒF2 on the conjugation parameter. The present paper investigates the existence of faithful, or non-faithful, solutions of similar families of quadratic equations corresponding to maps of absolute degree 0. The existence results are proved by constructing solutions. The non-existence results are based on studying two equations in ZŒ  and in its quotient Q , respectively, which are derived from the original equation and are easier to work with, where is the fundamental group of the target surface, and Q is the quotient of the abelian group ZŒ n f1g by the system of relations g g 1 , g 2 n f1g . Unknown variables of the first and second derived equations belong to , ZŒ  , Q , while the parameters of these equations are the projections of the conjugation parameter to and Q , respectively. In terms of these projections, sufficient conditions for the existence, or non-existence, of solutions of the quadratic equations in F2 are obtained.