Finite quotients of 3-manifold groups (Q6560725)

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\Nabelian groups including \textit{D. R. Heath-Brown} [Invent. Math. 118, No. 2, 331--370 (1994; Zbl 0815.11032)] and \textit{E. Fouvry} and \textit{J. Klüners} [Lect. Notes Comput. Sci. 4076, 40--55 (2006; Zbl 1143.11352)]. In addition, the authors add [\textit{J. S. Ellenberg} et al., Ann. Math. (2) 183, No. 3, 729--786 (2016; Zbl 1342.14055); \textit{M. Lipnowski} et al., ``Cohen-Lenstra heuristics and bilinear pairings in the presence of roots of unity'', Preprint, \url{arXiv:2007.12533}; \textit{W. Wang} and \textit{M. M. Wood}, Comment. Math. Helv. 96, No. 2, 339--387 (2021; Zbl 1472.11284); \textit{M. Matchett Wood}, J. Am. Math. Soc. 30, No. 4, 915--958 (2017; Zbl 1366.05098)] for other theoretic applications of the moment problem for more general random abelian groups. For non-abelian groups, we can see [\textit{N. Boston} and \textit{M. M. Wood}, Compos. Math. 153, No. 7, 1372--1390 (2017; Zbl 1390.11093)].\N\NOne of the challenges in this paper is that the moments of \(\pi_1(M)\) are too large to determine a unique distribution in theory by applying [\textit{W. Sawin}, ``Identifying measures on non-abelian groups and modules by their moments via reduction to a local problem'', Preprint, \url{arXiv:2006.04934}]. In other words, there are several challenges to overcome. First, the authors look at a method that proves a nearly optimal results for the non-abelian group moment problem. Second, the authors confront cases in which the moments are of a size where uniqueness in the moment problem fails. In these cases, they leverage the information from Theorem 1.1 in this article, which consists of four brand new results. One of the results in the theorem says that if \(M\) is a closed, oriented \(3\)-manifold, and \(V\) is an irreducible representation of \(\pi_1(M)\) over a finite field \(\kappa\), then \(\hbox{dim }H_1(\pi_1(M),V)\) \(=\) \(\hbox{dim }H_1(\pi_1(M),V^{\vee})\). Another challenge is that their construction of the distribution from the moments requires infinite alternating sums of group cohomology of general finite groups. We need to organize and simplify these sums sufficiently to be able to prove analytic bounds on their growth and detect whether they are 0 or not.