Some quadratic equations in the free group of rank 2

Details Some quadratic equations in the free group of rank 2 Some quadratic equations in the free group of rank 2

2009-04-08

arxiv.org/abs/0904.1421v1

Geom. Topol. Monogr. 14 (2008) 219-294

Mathematics

Group Theory

This is the version published by Geometry & Topology Monographs on 29 April 2008

Scientific paper

10.2140/gtm.2008.14.219

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. The problem has been studied by the authors 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 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 self-intersection index mu: F_2 --> Z[F_2] 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.

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