Abstract commensurators of profinite groups

TL;DR

This paper systematically studies the abstract commensurators of profinite groups, exploring their properties, topological structures, and applications to understanding embeddings into simple topological groups.

Contribution

It introduces the concept of the abstract commensurator for profinite groups, analyzes its properties, and applies it to classify embeddings into simple topological groups.

Findings

Defined the abstract commensurator for profinite groups.

Constructed a simple topological group containing the pro-2 completion of the Grigorchuk group.

Showed limitations on embedding certain profinite groups into compactly generated simple groups.

Abstract

In this paper we initiate a systematic study of the abstract commensurators of profinite groups. The abstract commensurator of a profinite group $G$ is a group $Comm(G)$ which depends only on the commensurability class of $G$. We study various properties of $Comm(G)$; in particular, we find two natural ways to turn it into a topological group. We also use $Comm(G)$ to study topological groups which contain $G$ as an open subgroup (all such groups are totally disconnected and locally compact). For instance, we construct a topologically simple group which contains the pro-2 completion of the Grigorchuk group as an open subgroup. On the other hand, we show that some profinite groups cannot be embedded as open subgroups of compactly generated topologically simple groups. Several celebrated rigidity theorems, like Pink's analogue of Mostow's strong rigidity theorem for simple algebraic groups defined over local fields and the Neukirch-Uchida theorem, can be reformulated as structure theorems for the commensurators of certain profinite groups.