Finite Groups Generated in Low Real Codimension
Ivan Martino, Rahul Singh

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 of subspaces fixed by subgroups of a finite linear group . When is a reflection group, this arrangement is precisely the hyperplane reflection arrangement of . We generalize the notion of finite reflection groups. We say that a group is generated (resp. strictly generated) in codimension if it is generated by its elements that fix point-wise a subspace of codimension at most (resp. precisely ). If is generated in codimension two, we show that the intersection lattice of is atomic. We prove that the alternating subgroup of a reflection group is strictly generated in codimension two, moreover, the subspace arrangement of is the truncation at rank two of the reflection arrangement . Further, we compute the intersection…
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