Arithmetic quotients of the mapping class group

To every Q-irreducible representation r of a finite group H, there corresponds a simple factor A of Q[H] with an involution τ. To this pair (A,τ), we associate an arithmetic group Ω consisting of all (2g-2)×(2g-2) matrices over a natural order 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 Ω is a virtual quotient of the mapping class group Mod(Σ_g), i.e. a finite index subgroup of Ω is a quotient of a finite index subgroup of Mod(Σ_g). This shows that Mod (Σ_g) has a rich family of arithmetic quotients (and “Torelli subgroups”) for which the classical quotient Sp(2g, 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 Mod(Σ_g) which are defined over various (subfields of) cyclotomic fields and are of type (Formula Presented.) for arbitrarily large m.