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 [Topology 20 (1981) 133–145] using a surface method
and generalizing a result of Wicks [J. London Math. Soc. 37 (1962)
433–444]. Based on different techniques, the problem has been studied
by the authors [Manuscripta Math. 107 (2002) 311–341 and Atti
Sem. Mat. Fis. Univ. Modena 49 (2001) 339–400] 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 one-parameter
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 µ: F2→Z[F2] 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
Z[π] and in its quotient Q, respectively, which are
derived from the original equation and are easier to work with, where
π is the fundamental group of the target surface, and Q is the
quotient of the abelian group Z[π╲{1}] by the
system of relations g∼-g-1, g∈π╲{1}. Unknown
variables of the first and second derived equations belong to π,
Z[π], Q, while the parameters of these equations are
the projections of the conjugation parameter to π 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.
Keywords
free groups, quadratic equations in free
groups, surfaces, absolute degree, Nielsen coincidence
theory, group homology, presentation of groups