City products of right-angled buildings and their universal groups

TL;DR

This paper introduces city products of right-angled buildings, constructs their universal groups, and demonstrates that different buildings can have isomorphic universal groups, revealing new insights into their large-scale geometry.

Contribution

It defines city products of right-angled buildings, constructs their universal groups, and shows these groups can be isomorphic for non-isomorphic buildings, expanding understanding of their symmetry properties.

Findings

Universal groups of city products relate to component buildings and Coxeter diagrams.

Existence of non-isomorphic buildings with topologically isomorphic universal groups.

Generalization of previous examples of universal group isomorphisms.

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 $\Delta_1,\dots,\Delta_n$ are right-angled buildings, then we construct a new right-angled building $\Delta := \mathrm{cityproduct}_M(\Delta_1,\dots,\Delta_n)$. We can recover the buildings $\Delta_1,\dots,\Delta_n$ 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 $\Delta_1,\dots,\Delta_n$ 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 Be{\ss}mann.