Open subgroups of the automorphism group of a right-angled building

TL;DR

This paper characterizes proper open subgroups of automorphisms of right-angled buildings, introducing new concepts like firm elements and firmness to analyze their structure and fixed chambers.

Contribution

It introduces the notion of firm elements and firmness in right-angled Coxeter groups, providing new tools to understand automorphism subgroups of right-angled buildings.

Findings

Proper open subgroups are finite index closed stabilizers of residues.

Introduces the concept of firm elements and firmness for Coxeter group elements.

Determines chambers fixed by the fixator of a ball using new notions.

Abstract

We study the group of type-preserving automorphisms of a right-angled building, in particular when the building is locally finite. Our aim is to characterize the proper open subgroups as the finite index closed subgroups of the stabilizers of proper residues. One of the main tools is the new notion of firm elements in a right-angled Coxeter group, which are those elements for which the final letter in each reduced representation is the same. We also introduce the related notions of firmness for arbitrary elements of such a Coxeter group and $n$-flexibility of chambers in a right-angled building. These notions and their properties are used to determine the set of chambers fixed by the fixator of a ball. Our main result is obtained by combining these facts with ideas by Pierre-Emmanuel Caprace and Timoth\'ee Marquis in the context of Kac-Moody groups over finite fields, where we had to replace the notion of root groups by a new notion of root wing groups.