Is the right-angled building associated to a universal group unique?

TL;DR

This paper investigates whether universal groups linked to right-angled buildings are unique and explores their actions on different buildings of the same type, establishing isomorphisms between such groups.

Contribution

It demonstrates that universal groups associated with different right-angled buildings of the same type can be isomorphic as topological groups, answering a key structural question.

Findings

Universal groups can act chamber-transitively on different right-angled buildings.

Universal groups associated with different buildings can be topologically isomorphic.

The paper provides a construction for such isomorphisms.

Abstract

A universal group is a subgroup of the group of type preserving automorphisms of a right-angled building and hence associated to this building. A question is then if this universal group can act chamber-transitively and with compact open stabilisers on a different right-angled building of the same type. We answer this question and define two universal groups associated to different right-angled buildings which are isomorphic as topological groups.