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 $Δ_1,\dots,Δ_n$ are right-angled buildings, then we construct a new right-angled building $Δ:= \mathrm{cityproduct}_M(Δ_1,\dots,Δ_n)$. We can recover the buildings $Δ_1,\dots,Δ_n$ 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 $Δ_1,\dots,Δ_n$ 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.