Construction of 2-local finite groups of a type studied by Solomon and Benson

Abstract

A p-local finite group is an algebraic structure with a classifying space which has many of the properties of p-completed classifying spaces of finite groups. In this paper, we construct a family of 2-local finite groups, which are exotic in the following sense: they are based on certain fusion systems over the Sylow 2-subgroup of Spin_7(q) (q an odd prime power) shown by Solomon not to occur as the 2-fusion in any actual finite group. Thus, the resulting classifying spaces are not homotopy equivalent to the 2-completed classifying space of any finite group. As predicted by Benson, these classifying spaces are also very closely related to the Dwyer-Wilkerson space BDI(4). Erratum (11 Feb 2005). An error in the paper was pointed out to the authors by Andy Chernak. The error is corrected in the erratum at the end of version 2, which should be read alongside the paper.