Approximating Simple Locally Compact Groups by Their Dense Locally Compact Subgroups

The class script S sign 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 ϵ S that is, we consider the dense locally compact subgroups of groups G ϵ S. We identify a class ℝ of almost simple groups that properly contains script S sign 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 script S sign and establish various original results for ℝ that are also new for the subclass script S sign, notably concerning the structure of the local Sylow subgroups and the full automorphism group.