Approximating Simple Locally Compact Groups by Their Dense Locally Compact Subgroups. (16th January 2019)

• Record Type: Journal Article
• Authors: Caprace, Pierre-Emmanuel; Reid, Colin; Wesolek, Phillip
• Is Part Of: International mathematics research notices. Volume 2021:Number 7(2021)
• Page Start: 5037
• Page End: 5110
• Publication Date: 2019-01-16
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rny298
• Languages: English
• ISSNs: 1073-7928

Abstract

Abstract: The class $\mathscr{S}$ of totally disconnected locally compact (tdlc) groups that are non-discrete, compactly generated, and topologically simple contains many compelling examples. In recent years, a general theory for these groups, which studies the interaction between the compact open subgroups and the global structure, has emerged. In this article, we study the non-discrete tdlc groups $H$ that admit a continuous embedding with dense image into some $G\in \mathscr{S}$ ; that is, we consider the dense locally compact subgroups of groups $G\in \mathscr{S}$ . We identify a class $\mathscr{R}$ of almost simple groups that properly contains $\mathscr{S}$ and is moreover stable under passing to a non-discrete dense locally compact subgroup. We show that $\mathscr{R}$ enjoys many of the same properties previously obtained for $\mathscr{S}$ and establish various original results for $\mathscr{R}$ that are also new for the subclass $\mathscr{S}$, notably concerning the structure of the local Sylow subgroups and the full automorphism group.