Compact Hyperbolic Coxeter $n$-Polytopes with $n+3$ Facets

Abstract

We use methods of combinatorics of polytopes together with geometrical and computational ones to obtain the complete list of compact hyperbolic Coxeter $n$-polytopes with $n+3$ facets, $4\le n\le 7$. Combined with results of Esselmann this gives the classification of all compact hyperbolic Coxeter $n$-polytopes with $n+3$ facets, $n\ge 4$. Polytopes in dimensions $2$ and $3$ were classified by Poincaré and Andreev.