Monodromy of cyclic coverings of the projective line

Abstract

Abstract. We show that the image of the pure braid group under the monodromy action on the homology of a cyclic covering of degree d of the projective line is an arithmetic group provided the number of ramification points is sufficiently large compared to the degree d and the ramification degrees are co-prime to d.

Curves with prescribed symmetry and associated representations of mapping class groups

“…This implies that G has ≤ g(S G ) − 1 generators, but can be very much non-abelian. Venkataramana [23] proved that the answer is also yes when g(S G ) = 0, G abelian and such that the number of irregular orbits does not exceed half the order of G.…”

Finiteness of hyperelliptic and superelliptic curves with CM Jacobians

“…Note that the special subvarieties obtained this way are defined by products of unitary groups and symplectic groups. We mention also that the main result of [Ven14] implies that the monodromy group of these Z(m, N, a) are arithmetic subgroups in the corresponding Mumford-Tate groups up to central part under suitable constraints upon the local monodromy data. Since the fundamental group of a Shimura variety only differs from an arithmetic subgroup of the derived part of its Mumford-Tate group by a finite quotient, hence a general Z(m, N, a), which is of dimension N − 3, cannot be a Shimura subvariety, using a direct computation of the dimension of a Shimura variety from its Mumford-Tate group, cf.…”

Images of Quantum Representations of Mapping Class Groups and Dupont–guichardet–wigner Quasi-Homomorphisms

“…McMullen [40] addressed the question of the arithmeticity of Burau representations of braid groups at roots of unity and Venkataramana [46,47] solved it affirmatively in the case where the order of the root is bounded by twice the number of strands. Burau representations are particular examples of quantum representations in genus zero.…”

“…The main motivation of this paper is to obtain new information about the images of mapping class groups by quantum representations by analyzing their 2-cohomology. McMullen ( [40]) addressed the question of the arithmeticity of Burau representations of braid groups at roots of unity and Venkataramana ( [46,47]) solved it affirmatively in the case where the order of the root is bounded by twice the number of strands. Burau representations are particular examples of quantum representations in genus zero.…”