Quotients of buildings by groups acting freely on chambers

TL;DR

This paper introduces Weyl graphs, a combinatorial tool to model quotients of Tits buildings by free group actions, enabling new ways to construct and analyze buildings and their fundamental groups.

Contribution

It develops the theory of Weyl graphs, generalizes Tits's chamber systems, and provides methods for constructing buildings and their fundamental groups via these graphs.

Findings

Weyl graphs generalize Tits's chamber systems.

Covering theory for Weyl graphs allows building construction as universal covers.

A method for obtaining group presentations of fundamental groups of Weyl graphs.

Abstract

We introduce certain directed multigraphs with extra structure, called Weyl graphs, which model quotients of Tits buildings by type-preserving chamber-free group actions. Their advantage over complexes of groups, which are often used for the CAT(0) Davis realization of buildings, is that Weyl graphs exploit the ultimate combinatorial W-metric structure of buildings. Weyl graphs generalize Tits's chambers systems of type M by allowing 2-residues to be quotients of generalized polygons by flag-free group actions, and Weyl graphs are easily constructed by amalgamating such quotients. We develop covering theory of Weyl graphs, which can be used to construct buildings as universal covers. We describe a method of obtaining a group presentation of the fundamental group of a Weyl graph, which acts chamber-freely on the covering building. The theory developed here is part of a fully general theory which deals with not necessarily chamber-free actions and the stacky version of buildings.