Abelian quotients of subgroups of the mapping class group and higher Prym representations (Q2843982)

Let \(\Sigma_{g,n}^p\) be an oriented genus \(g\) surface with \(n\) boundary components and \(p\) punctures, and \(\mathrm{Mod}_{g,n}^p\) be the mapping class group of \(\Sigma_{g,n}^p\), whose elements are represented by orientation preserving homeomorphisms fixing the boundary and punctures pointwise. The following is the well-known conjecture by \textit{N. V. Ivanov} [Proceedings of Symposia in Pure Mathematics 74, 71--80 (2006; Zbl 1281.57011)]: Conjecture 1.1. For some \(g \geq 3\) and \(n,p \geq 0\), let \(\Gamma < \mathrm{Mod}_{g,n}^p\) be a finite-index subgroup. Then \(H_1(\Gamma ; \mathbb{Q}) = 0\).NEWLINENEWLINELet \(K < \pi_1 (\Sigma_{g,n}^p)\) be a finite index subgroup which satisfies \(f (K) = K\) for any \(f \in \mathrm{Aut}(\pi_1 (\Sigma_{g,n}^p))\). Let \(S\) be the finite cover of \(\Sigma_{g,n}^p\) corresponding \(K\), and \(B\) be the subspace of \(\mathrm{H}_1(S;\mathbb{Q})= \mathrm{H}_1 (K;\mathbb{Q})\) spanned by the homology classes of the boundary components of \(S\) and loops freely homotopic into the punctures of \(S\). Then \(\mathrm{Mod}_{g,n}^{p+1}\) acts on \(V_K = \mathrm{H}_1(K;\mathbb{Q})/B\). The linear presentation \(\mathrm{Mod}_{g,n}^p \to \mathrm{Aut}(V_K)\) is a higher Prym presentation. In this paper, the following conjecture is made: Conjecture 1.2. Fix \(g \geq 2\) and \(n,p\geq 0\). Let \(K<\pi_1(\Sigma_{g,n}^p)\) be as above. Then, for all nonzero vectors \(v \in V_K\), the \(\mathrm{Mod}_{g,n}^{p+1}\)-orbit of \(v\) by the higher Prym presentation is infinite.NEWLINENEWLINEFor abbreviation, let NVSZ\((g,n,p)\) stand for the assertion that Conjecture 1.1 holds for \(\Sigma_{g,n}^p\), NFO\((g,n,p)\) stand for the assertion that Conjecture 1.2 holds for \(\Sigma_{g,n}^p\), and NVSZ\((g)\) stand for the assertion that Conjecture 1.1 holds for \(\Sigma_{g,n}^p\) for any \(n,p \geq 0\). In this paper, mainly by analyzing the action of the mapping class group on the curve complex and the stabilizers of this action, it is shown that, for fixed \(g \geq 3\) and \(p \geq 0\), (1) NFO\((g-1,n+1,p)\) implies NVSZ\((g,n,p)\) for \(n \geq 0\), and (2) NVSZ\((g,n,p+1)\) implies NFO\((g,n,p)\) for \(n \geq 1\). From the above result, it is shown that, if NVSZ\((G)\) holds for some \(G \geq 3\), then NVSZ\((g)\) holds for all \(g \geq G\).