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This paper is about cohomology of mapping class groups from the perspectiveof arithmetic groups. For a closed surface $S$ of genus $g$, the mapping classgroup $Mod(S)$ admits a well-known arithmetic quotient $Mod(S)\rightarrowSp(2g, Z)$, under which the stable cohomology of $Sp(2g,Z)$ pulls back toalgebra generated by the odd MMM classes of $Mod(S)$. We extend this example toother arithmetic groups associated to mapping class groups and explore some ofthe consequences for surface bundles.  For $G=Z/mZ$ and for a regular $G$-cover $S\rightarrow S'$ (possiblybranched), a finite index subgroup $\Gamma< Mod( S')$ admits a homomorphism toan arithmetic group $Sp^G<Sp(2g,Z)$. The induced map on cohomology can beunderstood using index theory. To this end, we describe a families version ofthe $G$-index theorem for the signature operator and apply this to (i) compute$H^2(Sp^G;Q)\rightarrow H^2(\Gamma;Q)$, (ii) re-derive Hirzebruch's formula forsignature of a branched cover (in the case of a surface bundle), (iii) computeToledo invariants of surface group representations to $SU(p,q)$ arising fromAtiyah--Kodaira constructions, and (iv) describe how classes in $H^*(Sp^G;Q)$give equivariant cobordism invariants for surface bundles with a fiberwise $G$action, following Church--Farb--Thibault.

Characteristic classes of fiberwise branched surface bundles via arithmetic