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Nielsen equivalence in Fuchsian groups

Martin Lustig and Yoav Moriah

Algebraic & Geometric Topology 22 (2022) 189–226
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 γi 3. In particular, the well-known problematic cases where G has characteristic exponents γi = 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