Finite Groups Generated in Low Real Codimension

TL;DR

This paper investigates the structure of finite groups generated in low real codimension, focusing on their intersection lattices, and extends the concept of reflection groups to broader classes.

Contribution

It introduces the notion of groups generated in low codimension, characterizes their intersection lattices, and computes these lattices for finite subgroups of GL(3,R), including new examples.

Findings

The intersection lattice of groups generated in codimension two is atomic.

The alternating subgroup of a reflection group is strictly generated in codimension two.

Computed the intersection lattice for all finite subgroups of GL(3,R).

Abstract

We study the intersection lattice of the arrangement $\mathcal{A}^G$ 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 $k$). If $G$ is generated in codimension two, we show that the intersection lattice of $\mathcal{A}^G$ is atomic. We prove that the alternating subgroup $\mathsf{Alt}(W)$ of a reflection group $W$ is strictly generated in codimension two, moreover, the subspace arrangement of $\mathsf{Alt}(W)$ is the truncation at rank two of the reflection arrangement $\mathcal{A}^W$. Further, we compute the intersection lattice of all finite subgroups of $GL_3(\mathbb{R})$, 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.