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

中文翻译：

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映射类组的非虚拟可解子组具有非虚拟可解表示

令$ \ Sigma $为具有至少一个边界分量的有限类型的紧凑可定向曲面。令$ \ Gamma \ leq \ mathrm {Mod}（\ Sigma）$为几乎不可解的子组。我们通过证明存在存在的有限元同构表示$ \ rho $ \ mathrm {Mod}（\ Sigma）$来回答Lubotzky问题，从而使$ \ rho（\ Gamma）$实际上不可解。然后，我们应用Lubotzky和Meiri的结果表明，对于在这样的组上的任何随机行走，降落在幂或拓扑熵为0的元素上的概率在行走长度上均呈指数下降。