Non virtually solvable subgroups of mapping class groups have non virtually solvable representations

Let Sigma be a compact orientable surface of finite type with at least one boundary component. Let Gamma <= Mod(Sigma) be a non virtually solvable subgroup. We answer a question of Lubotzky by showing that there exists a finite dimensional homological representation rho of Mod(Sigma) such that rho(Gamma) is not virtually solvable. We then apply results of Lubotzky and Meiri to show that for any random walk on such a group the probability of landing on a power, or on an element with topological entropy 0 both decrease exponentially in the length of the walk.