#### Vol. 4, No. 8, 2010

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

### Michael Larsen

Vol. 4 (2010), No. 8, 1029–1038
##### Abstract

Given a fixed integer $n$, we consider closed subgroups $\mathsc{G}$ of ${GL}_{n}\left({ℤ}_{p}\right)$, where $p$ is sufficiently large in terms of $n$. Assuming that the identity component of the Zariski closure $G$ of $\mathsc{G}$ in ${GL}_{n,{ℚ}_{p}}$ does not admit any nontrivial torus as quotient group, we give a condition on the ($mod\phantom{\rule{1em}{0ex}}p$) reduction of $\mathsc{G}$ which guarantees that $\mathsc{G}$ is of bounded index in ${GL}_{n}\left({ℤ}_{p}\right)\cap G\left({ℚ}_{p}\right)$.

##### Keywords
exponentially generated, Nori's theorem, $p$-adic Lie group
Primary: 20G25
Secondary: 20G40