On right-angled reflection groups in hyperbolic spaces

Abstract

We show that the right-angled hyperbolic polyhedra of finite volume in the hyperbolic space H n may only exist if n ≤ 14. We also provide a family of such polyhedra of dimensions n = 3, 4,. .. , 8. We prove that for n = 3, 4 the members of this family have the minimal total number of hyperfaces and cusps among all hyperbolic right-angled polyhedra of the corresponding dimension. This fact is used in the proof of the main result.

Wild knots in higher dimensions as limit sets of Kleinian groups

“…In particular, an OBC is locally finite. One has the following theorem of Potyagailo and Vinberg (see [28]): This theorem has, for our purposes, the following relevant corollary: Proof. The OBC is obtained as follows: Consider the half-space model of…”

Infinitely many hyperbolic Coxeter groups through dimension 19

“…Compact examples are known to exist in H n for n Ä 4 and finite-volume ones for n Ä 8. See Potyagailo and Vinberg [12] for these examples and also for a proof that compact (resp. finite-volume) examples cannot exist for n > 4 (resp.…”

“…The overline has the same meaning as before. For n D 3; : : : ; 8 we use the n-dimensional right-angled polyhedron from [12]. For n D 6, 7 and 8 it has three disjoint doubling walls, so we can use it for Q.…”

On the optimality of the ideal right-angled 24–cell

“…Call the sequence F 1 , : : : , F m a .k;`/ circuit, k C`D m, if it comprises k co-dimension two faces and`ideal vertices shared by the facets. We complete the analysis carried out by Potyagailo and Vinberg [9] in the following way. where ij > 0 is the length of the common perpendicular between two disjoint support hyperplanes for F i and F j respectively.…”

“…Furthermore, using the results of Khovanskiȋ [7] and Nikulin [8], we obtain a new dimension bound for ideal rightangled hyperbolic polytopes. The case of right-angled hyperbolic polytopes with both proper and ideal vertices was considered before by Dufour [6] and by Potyagailo and Vinberg [9].…”