TY - JOUR
T1 - Quotients of buildings by groups acting freely on chambers
AU - Norledge, William
N1 - Funding Information:
We thank Alina Vdovina, Corneliu Hoffman, David Stewart, and an anonymous reviewer for useful suggestions. We thank Anne Thomas for useful discussions during an early stage of this project. This research was partly supported by grant TRT 015 from Templeton Religion Trust as part of the mathematical picture language project at Harvard University. We also thank Newcastle University for their support.
Publisher Copyright:
© 2021 Elsevier B.V.
PY - 2021/11
Y1 - 2021/11
N2 - 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 chamber systems of type M by allowing rank two 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 for 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.
AB - 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 chamber systems of type M by allowing rank two 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 for 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.
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U2 - 10.1016/j.jpaa.2021.106730
DO - 10.1016/j.jpaa.2021.106730
M3 - Article
AN - SCOPUS:85102884323
VL - 225
JO - Journal of Pure and Applied Algebra
JF - Journal of Pure and Applied Algebra
SN - 0022-4049
IS - 11
M1 - 106730
ER -