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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


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.


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


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

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