2017/10/19 by William Norledge, Norledge, William
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1710.06968
openalex publication_date 2017/10/19 · openalex created_date 2022/09/12 · openalex updated_date 2026/07/28
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.