For a given quadratic equation with any number of unknowns in any free
group F, with righthand 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 oneparameter
family of quadratic equations in the free group F_{2} 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
selfintersection index µ: F_{2}→Z[F_{2}] on the
conjugation parameter. The present paper investigates the existence
of faithful, or nonfaithful, solutions of similar families of
quadratic equations corresponding to maps of absolute degree 0. The
existence results are proved by constructing solutions. The
nonexistence 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 nonexistence, of solutions of the quadratic
equations in F_{2} are obtained.
