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

TL;DR

The paper proves that non virtually solvable subgroups of mapping class groups have finite-dimensional representations with non virtually solvable images, and shows exponential decay in certain properties for random walks on these groups.

Contribution

It demonstrates the existence of finite-dimensional homological representations with non virtually solvable images for non virtually solvable subgroups of mapping class groups.

Findings

Existence of non virtually solvable images under finite-dimensional representations.

Exponential decay in probability for random walks landing on powers or zero-entropy elements.

Application of Lubotzky and Meiri's results to mapping class groups.

Abstract

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