Finite quotients of 3-manifold groups

Will Sawin; Melanie Matchett Wood

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arXiv · 2203.01140

Finite quotients of 3-manifold groups

Will Sawin·Melanie Matchett Wood

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Abstract

For $G$ and $H_1,\dots, H_n$ finite groups, does there exist a $3$-manifold group with $G$ as a quotient but no $H_i$ as a quotient? We answer all such questions in terms of the group cohomology of finite groups. We prove non-existence with topological results generalizing the theory of semicharacteristics. To prove existence of 3-manifolds with certain finite quotients but not others, we use a probabilistic method, by first proving a formula for the distribution of the (profinite completion of) the fundamental group of a random 3-manifold in the Dunfield-Thurston model of random Heegaard splittings as the genus goes to infinity. We believe this is the first construction of a new distribution of random groups from its moments.

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BibTeX RIS

Will Sawin, Melanie Matchett Wood. 2022-03-02. Finite quotients of 3-manifold groups. https://arxiv.org/abs/2203.01140

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