Irreducible generating tuples of Fuchsian groups

TL;DR

This paper proves the uniqueness and irreducibility of certain generating tuples in Fuchsian groups, providing insights into Nielsen classes and applications to 3-manifold groups.

Contribution

It introduces a new method to establish the uniqueness of almost orbifold covers with rigid generating tuples in Fuchsian groups.

Findings

Almost orbifold covers with rigid generating tuples are unique up to equivalence.

Irreducible generating tuples are shown to be irreducible.

Application to fundamental groups of Haken Seifert manifolds confirms irreducibility of certain Heegaard splittings.

Abstract

L. Louder showed that any generating tuple of a surface group is Nielsen equivalent to a stabilized standard generating tuple i.e. $(a_1,\ldots ,a_k,1\ldots, 1)$ where $(a_1,\ldots ,a_k)$ is the standard generating tuple. This implies in particular that irreducible generating tuples, i.e. tuples that are not Nielsen equivalent to a tuple of the form $(g_1,\ldots ,g_k,1)$, are minimal. In a previous work the first author generalized Louder's ideas and showed that all irreducible and non-standard generating tuples of sufficiently large Fuchsian groups can be represented by so-called almost orbifold covers endowed with a rigid generating tuple. In the present paper a variation of the ideas from \cite{W2} is used to show that this almost orbifold cover with a rigid generating tuple is unique up to the appropriate equivalence. It is moreover shown that any such generating tuple is irreducible. This provides a way to exhibit many Nielsen classes of non-minimal irreducible generating tuples for Fuchsian groups. As an application we show that generating tuples of fundamental groups of Haken Seifert manifolds corresponding to irreducible horizontal Heegaard splittings are irreducible.