Complexes of Groups

Abstract

Roughly speaking, complexes of groups were introduced to describe actions of groups on simply connected simplicial complexes in terms of suitable local data on the quotient. They are natural generalizations of the concept of graphs of groups due to Bass and Serre. A technical problem arises from the fact the quotient of a simplicial complex by a simplicial group action will not be a simplicial complex in general. Indeed, even if the set-wise stabilizer of each simplex is equal to its point-wise stabilizer, faces of a given simplex might get identified. (For example, regard the real line as a one-dimensional complex with vertices at the integers and consider the action of ℤ by translations.) Because of this problem, it is more natural to work with polyhedral complexes.