Stacky Buildings

TL;DR

This paper introduces W-groupoids, a new structure generalizing buildings and related concepts, providing a framework for constructing complex geometric objects like exotic and hyperbolic buildings through amalgamation and covering theory.

Contribution

The paper defines W-groupoids, characterizes buildings as their simply connected forms, and develops a covering theory to construct diverse types of buildings, including exotic and hyperbolic ones.

Findings

W-groupoids generalize Bruhat decompositions and buildings.

Buildings are characterized as connected simply connected W-groupoids.

Covering theory allows construction of complex buildings via amalgamation.

Abstract

We introduce structures which model quotients of buildings by type-preserving group actions. These structures, which we call W-groupoids for W a Coxeter group, generalize Bruhat decompositions, chambers systems of type M, Tits amalgams, and buildings themselves. We define the fundamental group of a W-groupoid, and characterize buildings as connected simply connected W-groupoids. We give a brief outline of covering theory of W-groupoids, which produces buildings as universal covers equipped with an action of the fundamental group. The local-to-global theorem of Tits concerning spherical 3-resides allows for the construction of W-groupoids by amalgamating quotients of generalized polygons along groupoids. In this way, W-groupoids provide a powerful way to construct (lattices in) exotic, hyperbolic, and wild buildings.