Construction of 2-local finite groups of a type studied by Solomon and Benson
by Ran Levi and Bob Oliver
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 $BDI(4)$.