[2210.03611] Irreducible generator tuples of the Fuchsian group

[Submitted on 7 Oct 2022]

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Overview: L. Louder showed that any generation tuple of surface groups is the Nielsen equivalent of the stabilized standard generation tuple, i.e. $(a_1,\ldots ,a_k,1\ldots, 1)$ . where $(a_1,\ldots ,a_k)$ is the standard generated tuple. In particular, this means that the irreducible generative tuple, i.e. the non-Nielsen equivalent of the tuple of the form $(g_1,\ldots ,g_k,1)$ , is the smallest. In a previous work, the first author generalized his Louder’s idea that all irreducible and non-canonical generative tuples of a sufficiently large Fuchsian group are represented by so-called nearly orbifold covers given rigid-body generative tuples. showed what it can do.

In this paper, we use a variation of the idea from \cite{W2} to show that nearly orbifold covers with this rigid-body generation tuple are unique up to proper equivalence. Furthermore, it is also shown that such a generated tuple is irreducible. This provides a way to denote the many Nielsen classes of nonminimum irreducible generator tuples of the Fuchsian group.

As an application, we show that the generator tuple of the fundamental group of Haken Seifert manifolds corresponding to the irreducible horizontal Heegaard partition is irreducible.