Mapping class groups of surfaces of genus $\geq 3$ do not virtually surject to $\mathbb{Z}$

TL;DR

This paper proves Ivanov's conjecture that for surfaces of genus at least 3, finite-index subgroups of their mapping class groups do not virtually surject onto the integers, implying certain homological properties of related moduli spaces.

Contribution

It confirms Ivanov's conjecture for genus ≥ 3 surfaces and derives consequences for the homology of finite covers of moduli spaces.

Findings

Finite-index subgroups of mapping class groups of genus ≥ 3 surfaces do not virtually surject onto ℤ.

The first homology over ℚ of finite covers of moduli spaces of genus ≥ 3 is zero.

The result resolves a well-known conjecture in the theory of surface mapping class groups.

Abstract

We prove a well known conjecture of Nikolai Ivanov which states that if $X$ is a surface of genus $\geq 3$ (with any number of punctures and boundary components), $\rm{Mod}(X)$ is the mapping class group of $X$, and $K < \rm{Mod}(X)$ is a finite-index subgroup, then $K$ does not virtually surject to $\mathbb{Z}$. As a corollary of this we get that $H_1(Z; \mathbb{Q}) = 0$ whenever $Z$ is a finite cover of $\mathcal{M}_{g,n}$, the moduli space of complex algebraic curves of genus $g\geq 3$ with $n$ marked points.