BASIC NONARCHIMEDEAN JØRGENSEN THEORY

IF 0.5 4区数学 Q3 MATHEMATICS

Journal of the Australian Mathematical Society Pub Date : 2024-03-19 DOI:10.1017/s1446788724000028

MATTHEW CONDER, HARRIS LEUNG, JEROEN SCHILLEWAERT

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引用次数: 0

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. Algebra261(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.

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基本非拱顶约根森理论

我们证明了约根森不等式的非archimedean 类似形式，并用它推导出了几个代数收敛结果。作为应用，我们证明了 ${mathrm {SL}_2}(K)$ 的每个密集子群（其中 K 是 p-adic 域）都包含两个元素，它们生成 ${mathrm {SL}_2}(K)$ 的密集子群，这是 Breuillard 和 Gelander ['论李群的密集自由子群'，J. Algebra261(2) (2003)，448-467] 结果的特例。我们还列出了其他几个相关结果，这些结果为专家所熟知，但在文献中却不易找到；例如，我们证明了在非拱顶局部域 K 上 ${mathrm {SL}_2}(K)$ 的非元素子群是离散的，当且仅当它的每个双发电机子群都是离散的。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of the Australian Mathematical Society 数学-数学

CiteScore

1.70

自引率

0.00%

发文量

审稿时长

6 months

期刊介绍： The Journal of the Australian Mathematical Society is the oldest journal of the Society, and is well established in its coverage of all areas of pure mathematics and mathematical statistics. It seeks to publish original high-quality articles of moderate length that will attract wide interest. Papers are carefully reviewed, and those with good introductions explaining the meaning and value of the results are preferred. Published Bi-monthly Published for the Australian Mathematical Society