Nielsen equivalence in Fuchsian groups

Lustig, Martin; Moriah, Yoav

doi:10.2140/agt.2022.22.189

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Nielsen equivalence in Fuchsian groups

Martin Lustig and Yoav Moriah

Algebraic & Geometric Topology 22 (2022) 189–226

DOI: 10.2140/agt.2022.22.189

Abstract

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 nonconjugate cyclic subgroups of order $γ_{i} \geq 3$ . In particular, the well-known problematic cases where $G$ has characteristic exponents $γ_{i} = 2$ are not excluded.

We classify generating systems up to Nielsen equivalence; this notion is strongly related to Heegaard splittings of $3$ –manifolds. The results presented here 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.

Keywords

Nielsen equivalence, Fuchsian groups, Fox derivatives, generating systems, Heegaard splittings

Mathematical Subject Classification 2010

Primary: 20H10, 57M99

References

Publication

Received: 19 January 2020

Revised: 30 December 2020

Accepted: 31 January 2021

Published: 26 April 2022

Authors

Martin Lustig
Département de mathématiques, I2M Université Aix-Marseille III UMR 7373 Marseille France

Yoav Moriah
Department of Mathematics Technion Haifa Israel

Volume 22, issue 1 (2022)