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

Let $\mathrm{\Sigma }$ be a compact orientable surface of finite type with at least one boundary component. Let $\mathrm{\Gamma }\le \text{Mod}(\mathrm{\Sigma })$ be a non virtually solvable subgroup. We answer a question of Lubotzky by showing that there exists a finite dimensional homological representation $\rho$ of $\text{Mod}(\mathrm{\Sigma })$ such that $\rho (\mathrm{\Gamma })$ 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.