Automorphism groups of right-angled buildings: simplicity and local splittings

TL;DR

This paper proves that automorphism groups of certain right-angled buildings are simple and explores their local structure, providing new examples of locally compact simple groups with interesting subgroup decompositions.

Contribution

It establishes the simplicity of automorphism groups of irreducible semi-regular thick right-angled buildings and analyzes their local subgroup structures.

Findings

Automorphism groups are abstractly simple.

In locally finite cases, they form a large family of simple locally compact groups.

Examples include groups that are locally indecomposable but have non-trivially decomposing local normal subgroups.

Abstract

We show that the group of type-preserving automorphisms of any irreducible semi-regular thick right-angled building is abstractly simple. When the building is locally finite, this gives a large family of compactly generated (abstractly) simple locally compact groups. Specializing to appropriate cases, we obtain examples of such simple groups that are locally indecomposable, but have locally normal subgroups decomposing non-trivially as direct products.