Approximating Simple Locally Compact Groups by Their Dense Locally Compact Subgroups

The class 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 is an element of S; that is, we consider the dense locally compact subgroups of groups G is an element of v. We identify a class R of almost simple groups that properly contains S and is moreover stable under passing to a non-discrete dense locally compact subgroup. We show that R enjoys many of the same properties previously obtained for S and establish various original results for R that are also new for the subclass S, notably concerning the structure of the local Sylow subgroups and the full automorphism group.