#### Volume 22, issue 1 (2022)

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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 ${\gamma }_{i}\ge 3$. In particular, the well-known problematic cases where $G$ has characteristic exponents ${\gamma }_{i}=2$ 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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