Finite groups generated in low real codimension

Abstract

We study the intersection lattice of the arrangement AG of subspaces fixed by subgroups of a finite linear group G. When G is a reflection group, this arrangement is precisely the hyperplane reflection arrangement of G. We generalize the notion of finite reflection groups. We say that a group G is generated (resp. strictly generated) in codimension k if it is generated by its elements that fix point-wise a subspace of codimension at most k (resp. precisely k).We prove that the alternating subgroup Alt(W) of a reflection group W is strictly generated in codimension two. Further, we compute the intersection lattice of all finite subgroups of GL3(R), and moreover, we emphasize the groups that are “minimally generated in real codimension two”, i.e., groups that are strictly generated in codimension two but have no real reflection representations. We also provide several examples of groups generated in higher codimension.