Incoherence of free-by-free and surface-by-free groups

Robert Kropholler; Stefano Vidussi; Genevieve Walsh

Incoherence of free-by-free and surface-by-free groups

Group Theory 2021-10-05 v3 Geometric Topology

Authors: Robert Kropholler , Stefano Vidussi , Genevieve Walsh

Abstract

Let $G$ be the semidirect product $\Gamma \rtimes F_2$ where $\Gamma$ is either the free group $F_n$ , $n > 1$ or the fundamental group $S_g$ of a closed surface of genus $g > 1$ . We prove that $G$ is incoherent, solving two problems posed by D. Wise. This implies an affirmative answer to a question of J. Hillman on the fundamental group of a surface bundle over a surface. Although many groups have been shown to be incoherent using virtual algebraic fibering, we also show that not every free-by-free group virtually algebraically fibers.

Keywords

incoherence free-by-free group surface-by-free group semidirect product virtual algebraic fibering fundamental group

Cite

@article{arxiv.2005.01202,
  title  = {Incoherence of free-by-free and surface-by-free groups},
  author = {Robert Kropholler and Stefano Vidussi and Genevieve Walsh},
  journal= {arXiv preprint arXiv:2005.01202},
  year   = {2021}
}

Comments

There is an error in the proofs of Theorem 3.1 and Theorem 3.5 which we cannot immediately fix. We thank Hongbin Sun for pointing this error out. We are merging the results on groups which do not virtually algebraically fiber (which are unaffected by this error) with our paper arXiv:2103.06930

Incoherence of free-by-free and surface-by-free groups

Abstract

Keywords

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