Arithmetic quotients of the mapping class group

Abstract

To every ${\mathbb{Q}}$-irreducible representation r of a finite group H, there corresponds a simple factor A of ${\mathbb{Q}[H]}$ with an involution ${\tau}$. To this pair ${(A, \tau)}$, we associate an arithmetic group ${\Omega}$ consisting of all ${(2g-2) \times (2g-2)}$ matrices over a natural order ${\mathfrak{O}^{op}}$ of ${A^{op}}$ which preserve a natural skew-Hermitian sesquilinear form on ${A^{2g-2}}$. We show that if H is generated by less than g elements, then ${\Omega}$ is a virtual quotient of the mapping class group ${{\rm Mod}(\Sigma_g)}$, i.e. a finite index subgroup of ${\Omega}$ is a quotient of a finite index subgroup of ${{\rm Mod}(\Sigma_g)}$. This shows that ${{\rm Mod}(\Sigma_g)}$ has a rich family of arithmetic quotients (and “Torelli subgroups”) for which the classical quotient ${{\rm Sp}(2g, \mathbb{Z})}$ is just a first case in a list, the case corresponding to the trivial group H and the trivial representation. Other pairs of H and r give rise to many new arithmetic quotients of ${{\rm Mod}(\Sigma_g)}$ which are defined over various (subfields of) cyclotomic fields and are of type ${{\rm Sp}(2m), {\rm SO}(2m, 2m),}$ and ${{\rm SU}(m, m)}$ for arbitrarily large m.