All finite groups are involved in the mapping class group
type: Item
instance of: scholarly article
title: All finite groups are involved in the mapping class group
zbMATH Open document ID: 1254.57018
author: Gregor Masbaum
author: Alan W. Reid
published in: Geometry \& Topology
publication date: 2012-08-20T00:00:00Z
full work available at URL: https://arxiv.org/abs/1106.4261
reviewed by: Susumu Hirose
Mathematics Subject Classification ID: 57M99
Mathematics Subject Classification ID: 57M07
Mathematics Subject Classification ID: 20F38
Mathematics Subject Classification ID: 57R56
zbMATH DE Number: 6068628
zbMATH Keywords: mapping class group
zbMATH Keywords: finite group
zbMATH Keywords: involved
zbMATH Keywords: unitary representation
zbMATH Keywords: Topological Quantum Field Theory
zbMATH Keywords: TQFT
Wikidata QID: Q57254102
MaRDI profile type: Publication
OpenAlex ID: W3105123864
arXiv ID: 1106.4261
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DOI: 10.2140/GT.2012.16.1393
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review text: Let \(\Gamma_g\) be the (orientation preserving) mapping class group of the closed orientable surface of genus \(g\). A group \(H\) is involved in a group \(G\) if there exists a finite index subgroup \(K\) of \(G\) and an epimorphism from \(K\) onto \(H\). The main theorem (Theorem 1.1) of this paper is that every finite group is involved in \(\Gamma_g\) for all \(g \geq 1\).NEWLINENEWLINEIn order to prove this theorem, it is shown (Theorem 1.2) that, for each \(g \geq 3\), there exist infinitely many \(N\) such that there exist infinitely many primes \(q\) such that \(\Gamma\) surjects onto \(PSL(N, \mathbb{F}_q)\), where \(\mathbb{F}_q\) is a finite field of order \(q\), and it is shown (Lemma 4.4) that, for any finite group \(H\), there exists an integer \(N\) such that for all odd primes \(q\), \(H\) is isomorphic to a subgroup of \(PSL(N, \mathbb{F}_q)\). For the proof of Theorem 1.2, the unitary representation of a central extension \(\tilde{\Gamma}_g\) of \(\Gamma_g\) by \(\mathbb{Z}\) arising in the \(SO(3)\)-TQFT constructed by \textit{C. Blanchet, N. Habegger, G. Masbaum} and \textit{P. Vogel} [Topology 34, No.4, 883--927 (1995; Zbl 0887.57009)] is used. The analogous result (Theorem 4.5) to Theorem 1.2 is also shown for the Torelli group.
description: scientific article; zbMATH DE number 6068628