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BASIC NONARCHIMEDEAN JØRGENSEN THEORY
Published online by Cambridge University Press: 19 March 2024
Abstract
We prove a nonarchimedean analogue of Jørgensen’s inequality, and use it to deduce several algebraic convergence results. As an application, we show that every dense subgroup of ${\mathrm {SL}_2}(K)$, where K is a p-adic field, contains two elements that generate a dense subgroup of ${\mathrm {SL}_2}(K)$, which is a special case of a result by Breuillard and Gelander [‘On dense free subgroups of Lie groups’, J. Algebra 261(2) (2003), 448–467]. We also list several other related results, which are well known to experts, but not easy to locate in the literature; for example, we show that a nonelementary subgroup of ${\mathrm {SL}_2}(K)$ over a nonarchimedean local field K is discrete if and only if each of its two-generator subgroups is discrete.
MSC classification
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- Research Article
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- Copyright
- © The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.
Footnotes
Communicated by George Willis
The first and third author are supported by the New Zealand Marsden Fund. The first author is also supported by the Rutherford Foundation.