Synthetic buildings for finite groups

Abstract

We define a synthetic building for a finite group G at the prime p to be a G-simplicial complex ∆ such that the fixed points under the action of any non-identity p-subgroup is contractible and such that every simplex stabilizer has order divisible by p. This concept is a generalization to all finite groups of the building of a finite group of Lie type, and also of the simplicial complex of non-identity p-subgroups of G. A major use of synthetic buildings is in the computation of the group cohomology of a finite group in terms of the cohomology of its subgroups, and at a more fundamental level they encode structural properties of the group. Aside from the p-subgroup complexes (which include classical buildings) and some sporadic geometries such as the C3-geometry for A7, there are few existing examples known of synthetic buildings. The goal of this dissertation is to classify them where possible, and at least to classify those of certain kinds. In the course of this we develop the general properties of synthetic buildings and identify significant desirable conditions that they may satisfy. We present new examples of synthetic buildings and provide the architecture to construct more. We focus primarily on symmetric, alternating, and Mathieu groups, and indeed find new synthetic buildings in all three cases. Furthermore, we provide the means for generating an infinite family of synthetic buildings for A5 at p = 2 in which every A5-complex is of distinct homotopy type. In the case when G has a strongly p-embedded subgroup, we prove that G has a 0-dimensional synthetic building at p; if the subgroups of G satisfy further conditions, we can guarantee the existence of infinitely many synthetic buildings for G at p of distinct homotopy types and bounded dimension, a tantalizing phenomenon that has not been observed before.