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Nielsen equivalence in Fuchsian groups

Martin Lustig and Yoav Moriah

Algebraic & Geometric Topology 22 (2022) 189–226

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.

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Nielsen equivalence, Fuchsian groups, Fox derivatives, generating systems, Heegaard splittings
Mathematical Subject Classification 2010
Primary: 20H10, 57M99
Received: 19 January 2020
Revised: 30 December 2020
Accepted: 31 January 2021
Published: 26 April 2022
Martin Lustig
Département de mathématiques, I2M
Université Aix-Marseille III
UMR 7373
Yoav Moriah
Department of Mathematics