Essential hyperbolic Coxeter polytopes

Abstract

We introduce a notion of essential hyperbolic Coxeter polytope as a polytope which fits some minimality conditions. The problem of classification of hyperbolic reflection groups can be easily reduced to classification of essential Coxeter polytopes. We determine a potentially large combinatorial class of polytopes containing, in particular, all the compact hyperbolic Coxeter polytopes of dimension at least 6 which are known to be essential, and prove that this class contains finitely many polytopes only. We also construct an effective algorithm of classifying polytopes from this class, realize it in four-dimensional case, and formulate a conjecture on finiteness of the number of essential polytopes.Research was partially supported by grants RFBR 07-01-00390-a and RFBR 11-01-00289-a.

All hyperbolic Coxeter n-cubes

“…and Σ 5 are arithmetic.Remark 10Similarly to the 5-dimensional case (seeRemark 7), one can show that the arithmetic non-cocompact Coxeter group with Coxeter graph Σ 5 is commensurable to the Coxeter simplex group with Coxeter symbol[3,4,3,4].…”

“…For example, hyperbolic Coxeter simplices exist only for 2 ≤ n ≤ 9. An overview of the current knowledge about the classification of hyperbolic Coxeter polyhedra (and related questions) is available on Anna Felikson's webpage [3], for example. Hyperbolic n-cubes are simple polyhedra bounded by 2n facets in H n , and, unlike simplices, they have no simplex facet.…”

Near Classification of Compact Hyperbolic Coxeter $d$-Polytopes with $d+4$ Facets and Related Dimension Bounds

“…This can be strengthened in dimension 4. Felikson and Tumarkin [14] later studied d polytopes with n facets having at most n − d − 2 pairs of disjoint facets. They gave a finite algorithm for listing these polytopes, and carried out this algorithm in dimension 4.…”

“…This includes the first known Coxeter polytope in dimension > 3 with an angle of less than π 10 and the first known Coxeter polytope in dimension > 3 with an angle of π 7 , along with many new essential polytopes. A polytope is essential if it is minimal with respect to the operations of taking the fundamental domain of a finite index reflection subgroup of the corresponding reflection group, or gluing two Coxeter polytopes along congruent facets (see [14] for further details). The present work combined with that of Felikson and Tumarkin yields a classification in all dimensions except 6, where the only known polytope was constructed by Bugaenko [6].…”

Arithmetic hyperbolic reflection groups

“…In a series of papers Felikson and Tumarkin studied other types of the hyperbolic Coxeter polyhedra (without connection to arithmeticity). We refer to [FT14] and the references therein for the related results.…”