Some quadratic equations in the free group of rank 2

TL;DR

This paper investigates the existence of solutions to parametric families of quadratic equations in free groups, focusing on cases related to maps between closed surfaces with specific degrees, extending previous work on degree 2 cases.

Contribution

The paper extends the analysis of quadratic equations in free groups to the case of maps with absolute degree 0, providing criteria for the existence of solutions based on algebraic invariants.

Findings

Solved the existence problem for degree 0 maps in terms of algebraic parameters

Extended previous degree 2 results to degree 0 cases

Provided conditions for faithful and non-faithful solutions

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. 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.