All finite groups are involved in the mapping class group (Q441120)

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.