Spherical simplices generating discrete reflection groups

TL;DR

This paper develops criteria to identify when a spherical simplex generates a discrete reflection group and classifies all indecomposable such simplices.

Contribution

It introduces new criteria for when spherical simplices generate discrete reflection groups and classifies all indecomposable cases.

Findings

Established criteria for discreteness of reflection groups

Classified all indecomposable spherical simplices generating such groups

Provided a comprehensive description of these simplices

Abstract

Let $P$ be a simplex in $S^n$ and $G_P$ be a group generated by the reflections with respect to the facets of $P$. We are interested in the case when the group $G_P$ is discrete. In this case we say that $G$ generates the discrete reflection group $G_P$. We develop the criteria for a simplex generating a discrete reflection group. We also describe all indecomposable spherical simplices generating discrete reflection groups.