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.

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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)