Geometry & Topology Monographs 14 (2008) 219–294

DOI: 10.2140/gtm.2008.14.219

arXiv: 0904.1421

Abstract

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

Mathematical Subject Classification

Primary: 20E05, 20F99

Secondary: 20F05, 55M20, 57M07

References

Publication

Received: 31 July 2006
Revised: 24 April 2008
Accepted: 14 February 2007
Published: 29 April 2008

Authors