Volume 14 (2008)

Download this article
For screen
For printing
Recent Volumes
Volume 1, 1998
Volume 2, 1999
Volume 3, 2000
Volume 4, 2002
Volume 5, 2002
Volume 6, 2003
Volume 7, 2004
Volume 8, 2006
Volume 9, 2006
Volume 10, 2007
Volume 11, 2007
Volume 12, 2007
Volume 13, 2008
Volume 14, 2008
Volume 15, 2008
Volume 16, 2009
Volume 17, 2011
Volume 18, 2012
Volume 19, 2015
The Series
MSP Books and Monographs
About this Series
Editorial Board
Ethics Statement
Author Index
Submission Guidelines
Author Copyright Form
Purchases
ISSN (electronic): 1464-8997
ISSN (print): 1464-8989
Other MSP Publications

Some quadratic equations in the free group of rank 2

Daciberg Gonçalves, Elena Kudryavtseva and Heiner Zieschang

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: F2Z[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
Daciberg Gonçalves
Departamento de Matemática
IME-USP
Caixa Postal 66281
Agência Cidade de São Paulo
05314-970 São Paulo SP
Brasil
Elena Kudryavtseva
Department of Mathematics and Mechanics
Moscow State University
Moscow 119992
Russia
Heiner Zieschang
Fakultät für Mathematik
Ruhr-Universität Bochum
44780 Bochum
Germany