Exponential generation and largeness for compact p-adic Lie groups

Larsen, Michael

doi:10.2140/ant.2010.4.1029

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

Michael Larsen

Vol. 4 (2010), No. 8, 1029–1038

DOI: 10.2140/ant.2010.4.1029

Abstract

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

Keywords

exponentially generated, Nori's theorem, $p$-adic Lie group

Mathematical Subject Classification 2000

Primary: 20G25

Secondary: 20G40

Milestones

Received: 15 May 2009

Revised: 21 July 2010

Accepted: 21 July 2010

Published: 24 February 2011

Authors

Michael Larsen
Department of Mathematics Rawles Hall Indiana University Bloomington, IN 47405-5701 United States
http://mlarsen.math.indiana.edu/~larsen/

Vol. 4, No. 8, 2010