Spherical simplices generating discrete reflection groups

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.