City products of right-angled buildings and their universal groups

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 Delta(1), ... , Delta(n) are right-angled buildings, then we construct a new right-angled building Delta := (sic)(M) (Delta(1) , . . . , Delta(n)). We can recover the buildings Delta(1) , . . . , Delta(n) as residues of Delta, but we can also construct a skeletal building of type M from Delta that captures the large-scale geometry of Delta. We then proceed to study universal groups for city products of right-angled buildings, and we show that the universal group of Delta can be expressed in terms of the universal groups for the buildings Delta(1) , . . . , Delta(n) 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. (c) 2022 Elsevier Inc. All rights reserved. D4