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

Abstract

Let $Σ$ be a compact orientable surface of finite type with at least one boundary component. Let $Γ\leq \textup{Mod}(Σ)$ be a non virtually solvable subgroup. We answer a question of Lubotzky by showing that there exists a finite dimensional homological representation $ρ$ of $\textup{Mod}(Σ)$ such that $ρ(Γ)$ 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.