Exponential generation and largeness for compact $p$-adic Lie groups

Abstract

Given a fixed integer $n$, we consider closed subgroups $\mathcal{G}$ of ${GL}_n\left({\mathbb{Z}}_p\right)$, where $p$ is sufficiently large in terms of $n$. Assuming that the identity component of the Zariski closure $G$ of $\mathcal{G}$ in ${GL}_{n,{\mathbb{Q}}_p}$ does not admit any nontrivial torus as quotient group, we give a condition on the ($modp$) reduction of $\mathcal{G}$ which guarantees that $\mathcal{G}$ is of bounded index in ${GL}_n\left({\mathbb{Z}}_p\right)\cap G\left({\mathbb{Q}}_p\right)$.