On finite index reflection subgroups of discrete reflection groups

TL;DR

This paper proves that for a discrete reflection group with a finite volume fundamental chamber, any finite index reflection subgroup has a fundamental chamber with at least as many facets, establishing a lower bound on its complexity.

Contribution

It establishes a lower bound on the number of facets of the fundamental chamber of finite index reflection subgroups in discrete reflection groups.

Findings

The fundamental chamber of the subgroup has at least as many facets as that of the original group.

The result applies to groups generated by reflections in hyperbolic or Euclidean space.

Provides a geometric constraint on the structure of finite index reflection subgroups.

Abstract

Let $G$ be a discrete group generated by reflections in hyperbolic or Euclidean space, and $H\subset G$ be a finite index subgroup generated by reflections. Suppose that the fundamental chamber of $G$ is a finite volume polytope with $k$ facets. In this paper, we prove that the fundamental chamber of $H$ has at least $k$ facets.