Unramified correspondences and virtual properties of mapping class groups

Abstract

We establish a connection between the conjecture of Bogomolov–Tschinkel about unramified correspondences and the Ivanov conjecture about the virtual homology of mapping class groups. Given g ⩾ 2 $g\geqslant 2$ , we show that every genus g $g$ Riemann surface X $X$ virtually dominates a fixed Riemann surface Y $Y$ of genus at least two if and only if there exists a finite index subgroup Γ < Mod g 1 $\Gamma <\operatorname{Mod}^1_g$ which allows a point pushing epimorphism onto a free group of rank two. As a consequence of this result, we show that the Putman–Wieland conjecture about the Higher Prym representations does not hold when g = 2 $g=2$ .