City products of right-angled buildings and their universal groups

Abstract

We introduce the notion of city products of right-angled buildings that produces a new right-angled building out of smaller ones. More precisely, if M is a right-angled Coxeter diagram of rank n and Δ1,…,Δn are right-angled buildings, then we construct a new right-angled building ▪. We can recover the buildings Δ1,…,Δn as residues of Δ, but we can also construct a skeletal building of type M from Δ that captures the large-scale geometry of Δ.We then proceed to study universal groups for city products of right-angled buildings, and we show that the universal group of Δ can be expressed in terms of the universal groups for the buildings Δ1,…,Δn and the structure of M. As an application, we show the existence of many examples of pairs of different buildings of the same type that admit (topologically) isomorphic universal groups, thereby vastly generalizing a recent example by Lara Beßmann.