Nielsen Equivalence in Fuchsian groups

TL;DR

This paper classifies minimal generating systems in a broad class of Fuchsian groups up to Nielsen equivalence, providing tools for understanding Heegaard splittings in 3-manifolds and extending previous work in the field.

Contribution

It offers a complete classification of generating systems in certain Fuchsian groups up to Nielsen equivalence, including cases with characteristic exponents equal to 2.

Findings

Classified minimal generating systems in a broad class of Fuchsian groups

Included groups with at least seven non-conjugate cyclic subgroups of order > 2

Provided tools for extending classification of Heegaard splittings in 3-manifolds

Abstract

In this paper we give a complete classification of minimal generating systems in a very general class of Fuchsian groups G. This class includes for example any G which has at least seven non-conjugate cyclic subgroups of order greater than 2. In particular, the well known problematic cases where G has characteristic exponents equal to 2 are not excluded. We classify generating systems up to Nielsen equivalence; this notion is strongly related to Heegaard splittings of 3-manifolds. The results of this paper provide in particular the tools for a rather general extension of previous work of the authors and others, on the isotopy classification of such splittings in Seifert fibered spaces.