Finite quotients of 3-manifold groups
type: Item
instance of: scholarly article
title: Finite quotients of 3-manifold groups
zbMATH Open document ID: 1545.57016
author: William F. Sawin
author: Melanie Matchett Wood
published in: Inventiones Mathematicae
publication date: 2024-06-23T00:00:00Z
review text: In this paper, the authors discuss the question of what finite quotients in which combinations the fundamental group of a \(3\)-manifold can have and not have.\N\NA fact is that if \(G\) is any finite group, then we can realize a closed 3-manifold \(M\) such that \(G\) is a quotient of \(\pi_1(M)\). On one hand, we may ask questions about all possible finite quotients of \(3\)-manifold groups. More specifically, we wonder, given finite groups \(G\) and \(H_1,\dots,H_n\), whether there exists a closed \(3\)-manifold \(M\) such that \(G\) is a quotient but no \(H_i\) is a quotient. In this article, the authors give an answer to such questions in terms of the cohomology of finite groups.\N\NFirst, the authors prove a theorem which provides certain obstructions to the existence of \(3\)-manifold groups with certain quotients but not others. Then, they show that when these obstructions vanish, not only do such 3-manifolds exist, but partially have quotients and non-quotients as desired.\N\NI felt quite interested in reading a section regarding comparison between previous work and new approaches. The authors note that the motivation for their work is due to Dunfield and Thurston's introduction of the model of random Heegaard splittings in [\textit{N. M. Dunfield} and \textit{W. P. Thurston}, Invent. Math. 166, No. 3, 457--521 (2006; Zbl 1111.57013)]. Dunfield and Thurston proved several results on this model, for example, the limiting probability of a manifold having positive first Betti number is \(0\).\N\NMoreover, the proof of Theorem 6.21 in [loc. cit.] on the average of \(\sharp \mathrm{Surj}(\pi_1(M),Q)\), what the authors call the \textit{moments} of the random group \(\pi_1(M)\), is a key input discussed in this article here (See Theorem 1.2). In this context, the task to prove Theorem 1.2 is to show that these averages actually determine entirely the distribution of random groups. There is a significant history of work on this \textit{moment problem} for random\N…
reviewed by: Ryo Ohashi
Mathematics Subject Classification ID: 57M10
Mathematics Subject Classification ID: 57M60
Mathematics Subject Classification ID: 20J06
Mathematics Subject Classification ID: 57K30
Mathematics Subject Classification ID: 57M05
Mathematics Subject Classification ID: 57M07
zbMATH DE Number: 7870149
zbMATH Keywords: finite group
zbMATH Keywords: fundamental group
zbMATH Keywords: 3-manifold
zbMATH Keywords: quotient
zbMATH Keywords: cohomology of groups
MaRDI profile type: Publication
cites work: FREE FINITE GROUP ACTIONS ON RATIONAL HOMOLOGY 3-SPHERES
cites work: The virtual Haken conjecture (with an appendix by Ian Agol, Daniel Groves and Jason Manning).
cites work: 3-manifold groups
cites work: Q3288660
cites work: Non-abelian Cohen–Lenstra heuristics over function fields
cites work: Dense embeddings of surface groups
cites work: Q3708977
cites work: On a Cohen-Lenstra heuristic for Jacobians of random graphs
cites work: A Note on Jacobians, Tutte Polynomials, and Two-Variable Zeta Functions of Graphs
cites work: Q3222942
cites work: Q3476914
cites work: Differentiable periodic maps
cites work: Free actions of finite groups on rational homology 3-spheres
cites work: Q5340151
cites work: Semicharacteristics, Bordism, and Free Group Actions
cites work: Non-triviality of the \(A\)-polynomial for knots in \(S^3\)
cites work: Finite covers of random 3-manifolds
cites work: Quantum invariants of random 3-manifolds
cites work: Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields
cites work: Q3035404
cites work: Arithmetic quotients of the mapping class group
cites work: The Weil representation in characteristic two
cites work: Small eigenvalues of random 3-manifolds
cites work: The size of Selmer groups for the congruent number problem. II. With an appendix by P. Monsky.
cites work: Automorphisms of group extensions and differentials in the Lyndon-Hochschild-Serre spectral sequence
cites work: Q3503664
cites work: Semicharacteristic classes
cites work: The free group on \(n\) generators modulo \(n + u\) random relations as \(n\) goes to infinity
cites work: A predicted distribution for Galois groups of maximal unramified extensions
cites work: Random methods in 3-manifold theory
cites work: Random Heegaard splittings
cites work: Cohen-Lenstra heuristic and roots of unity
cites work: The distribution of sandpile groups of random regular graphs
cites work: Groups Which Act on S n Without Fixed Point
cites work: Q4935965
cites work: Varieties of groups
cites work: On CW-complexes over groups with periodic cohomology
cites work: Mod 2 semi-characteristics and the converse to a theorem of Milnor
cites work: A model for random three-manifolds
cites work: Profinite Groups
cites work: Volumes of random 3‐manifolds
cites work: Moments and interpretations of the Cohen-Lenstra-Martinet heuristics
cites work: The distribution of sandpile groups of random graphs
cites work: Random integral matrices and the Cohen-Lenstra heuristics
DOI: 10.1007/S00222-024-01262-4
description: scientific article; zbMATH DE number 7870149