Hyperbolic Coxeter polytopes

Disclaimer:

This is an attempt to collect some results concerning classification and properties of hyperbolic Coxeter polytopes.
This page is under construction. Any corrections, suggestions or other comments are very welcome.

Arithmetic groups:

For a detailed discussion of advances in the arithmetic case see the recent survey by M. Belolipetsky [Bel].
See also the webpage on arithmetic hyperbolic reflection groups by Nikolay Bogachev.

Why "hyperbolic": spherical and Euclidean Coxeter polytopes are classified by H.S.M.Coxeter in 1934 [Cox].

Absence in large dimensions:

Compact hyperbolic Coxeter polytopes:

do not exist in dimensions dim>29 [Vin2];
examples are known only up to dim=8, the unique known example in dim=8 and both known examples in dim=7 are due to Bugaenko [Bug1].

Finite volume hyperbolic Coxeter polytopes:

do not exist in dimensions dim>995 [Pr1], [Khov];
examples are known in dimensions dim≤19 [Vin4], [KV] and dim=21 [Bor].

Some known classifications:

By dimension (dim):

dim=2: there exists a n-gon with given angles if and only if the sum of angles is less than π(n-2) [Po].
dim=3: see Andreev's theorem [And1], [And2], [RHD]. See also [Pog].

By number of facets (n):

n=dim+1: compact simplices (Lannér diagrams [Lan], dim=2,3,4) and non-compact simplices (quasi-Lannér diagrams [Ch], [Vin7], [Bou], dim=2,...,9).
n=dim+2:
- Products of two simplices:
  - Simplicial prisms exist in dim=3,4,5 [Kap], see also [Vin3].
  - Other products of two simplices (exist in dim=4 only): Esselmann polytopes [Ess1],[Ess2] and the unique non-compact polytope [Tum1].
- Pyramids over a product of two simplices [Tum1], dim=3,...,13, 17.
n=dim+3:
- Compact: exist in dim=2,...,6,8 only; see the list [Ess1], [Tum2]. First high-dimensional results are due to Bugaenko [Bug2].
- Finite volume:
  - do not exist in dim≥17 [Tum3], [Tum3'].
  - the unique polytope in dim=16 [Tum3], [Tum3'].
  - Pyramids over a product of three simplices [Tum3], dim=4,...,9,13.
  - polytopes with exactly one non-simple vertex exist in dim=4,...,10, see the list (see pp. 8-33) [Rob].
n=dim+4: compact polytopes with n=dim+4 facets do not exist in dim>7 [FT7]. There is a unique compact polytope in dim=7 with 11 facets [FT7] (constructed by Bugaenko [Bug2]).
- Dim=4 and 5: classification of compact hyperbolic Coxeter polytopes with dim+4 facets [Bur1], [Bur2] and [MZ1], [MZ2].

By number of pairs of non-intersecting facets (p):

p=0:
- compact: either a simplex, or an Esselmann polytope [FT1], see also [Pr2] for absence of such polytopes with angles π/2 and π/3;
- non-compact simple polytope: either a simplex or this polytope [FT1];
- non-compact polytopes with angles π/2 and π/3 [Pr2].
p=1: a compact polytope with p=1 satisfies n≤d+3 [FT2].
p≤n-dim-2: the number of compact polytopes in dim≥4 with p≤n-dim-2 is finite; all such polytopes can be listed by a finite algorithm [FT3].

By combinatorial type:

Pyramids:
- Pyramids over a product of two simplices [Tum1], dim=3,...,13, 17.
- Pyramids over a product of three simplices [Tum3], dim=4,...,9,13.
- Pyramids over products of more than three simplices: these are pyramids over products of 4 simplices, dim=5 [Mcl].
Cubes:
- Do not exist in dim≥10; ideal cubes only exist in dim=2,3 and are classified [Jacq].
- Complete classification of cubes (cubes only exist up to dim=5) [JT].
Products of simplices: is either a cube or a product of at most 3 simplices [Alex].

Right-angled polytopes:

Compact polytopes do not exist for dim>4, examples known for dim=2,3,4 [PV].
Finite volume polytopes do not exist for dim>14, examples only known for dim≤8 [PV].
Finite volume polytopes do not exist for dim>12 [Duf].
Estimate for the number of cusps of right-angled polytopes [Non].
Volumes of right-angled 3-polytopes: eleven smallest values [In], [Ves].

Polytopes obtained by gluings of smaller polytopes:

Examples obtained by gluings [Mak], [Ruz].
Infinitely many hyperbolic Coxeter groups through dimension 19 (all obtained via doublings of fundamental domain) [All].
Examples of non-arithmetic (non-cocompact) cofinite discrete reflection groups in n-dimensional hyperbolic space for all n≤12 and for n=14,18 [Vin5].

Volumes, subgroups, commensurability:

Volumes:

Volumes of hyperbolic simplices [JKRT1].
Volumes of compact 4-dimensional polytopes that are products of simplices [FTZ].
Volumes in even dimensions [Heck].

Subgroups:

A fundamental polytope of a subgroup has more faces than the fundamental polytope of a group [FT4].
Subgroup relations for hyperbolic simplices, [FT5].

Commensurability:

Commensurability of hyperbolic simplices [JKRT2], see also [FT5].
Commensurability of pyramids over a product of two simplices [GJK].

Other results:

Absence of simple ideal polytopes in dim>8 [FT6].

Rolling of Coxeter polyhedra along mirrors [AMN].

Bugaenko's 6-polytope with 34 facets: see p.6 in [All]

Polytopes defined by Napier cycles [ImH1], [ImH2].

Polytopes with mutually intersecting faces and dihedral angles π/2 and π/3: simplices and polytopes shown here [Pr2].

References:

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[AMN] D. Alekseevski, P. W. Michor, Y. A. Neretin, Rolling of Coxeter polyhedra along mirrors, Geometric Methods in Physics, XXXI Workshop Bialowieza, Poland, Birkhauser (2013), 67-86, arXiv:0907.3502.

[All] D. Allcock, Infinitely many hyperbolic Coxeter groups through dimension 19, Geom. Topol. 10 (2006), 737–758. arXiv:0903.0138.

[And1] E. M. Andreev, On convex polyhedra in Lobachevskii spaces, Math. USSR Sbornik 10 (1970), 413–440. Russian version.

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[Bel] M. Belolipetsky, Arithmetic hyperbolic reflection groups, Bull. (New Series) of the Am. Maths. Soc., Vol. 53, N 3 (2016), 437–475 arXiv:1506.03111.

[Bor] R. Borcherds, Automorphism groups of Lorentzian lattices, Journal of Algebra, Volume 111, Issue 1 (1987), 133-153.

[Bou] N. Bourbaki, Groupes et algebres de Lie, Ch. 4-6. Hermann, Paris (1968).

[Bug1] V. O. Bugaenko, Arithmetic crystallographic groups generated by reflections, and reflective hyperbolic lattices, Advances in Soviet Mathematics 8 (1992), 33--55.

[Bug2] V. O. Bugaenko, Groups of automorphisms of unimodular hyperbolic quadratic forms over the ring Z[(√ 5 +1)/2], Moscow Univ. Math. Bull. 39 (1984), 6–14, Russian version.

[Bur1] A. Burcroff, On compact hyperbolic Coxeter polytopes with few facets, MSc Thesis, Durham (2021).

[Bur2] A. Burcroff, Near Classification of Compact Hyperbolic Coxeter d-Polytopes with d+4 Facets and Related Dimension Bounds.

[Ch] M. Chein, Recherche des graphes des matrices des Coxeter hyperboliques d'ordre ≤10, ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 3 (1969) no. R3, 3-16.

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[FT3] A. Felikson, P. Tumarkin, Essential hyperbolic Coxeter polytopes, Israel Journal of Mathematics (2014), 199, 1, 113–161, arXiv:0906.4111.

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[FT5] A. Felikson, P. Tumarkin, Hyperbolic subalgebras of hyperbolic Kac-Moody algebras, Transform. Groups 17 (2012), 87-122 arXiv:1012.1046.

[FT6] A. Felikson, P. Tumarkin, On simple ideal hyperbolic Coxeter polytopes, Izv. Math. 72 (2008), 113-126, arXiv:math/0502413.

[FT7] A. Felikson, P. Tumarkin, On compact hyperbolic Coxeter d-polytopes with d+4 facets, Trans. Moscow Math. Soc. 69 (2008), 105-151, arXiv:math/0510238.

[FTZ] A. Felikson, P. Tumarkin, T. Zehrt, On hyperbolic Coxeter n-polytopes with n+2 facets, Adv. Geom. 7 (2007), 177-189.

[Jacq] M. Jacquemet, On hyperbolic Coxeter n-cubes Europ. J. Combin. 59 (2017), 192-203.

[JT] M. Jacquemet, S. Tschantz, All hyperbolic Coxeter n-cubes J. Combin. Theory A, 158 (2018), 387-406, arXiv:1803.10462.

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[JKRT2] N. Johnson, R. Kellerhals, J. Ratcliffe, S. Tschantz, Commensurability classes of hyperbolic Coxeter simplex reflection groups, Linear Algebra Appl. 345 (2002), 119-147.

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[ImH2] H.-Ch. Im Hof, Napier cycles and hyperbolic Coxeter groups, Bull. Soc. Math. de Belg. Serie A, XLII (1990), 523–545.

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[Lan] F. Lannér, On complexes with transitive groups of automorphisms, Comm. Sem. Math. Univ. Lund 11 (1950), 1–71.

[MZ1] J. Ma, F. Zheng Compact hyperbolic Coxeter 4-polytopes with 8 facets.

[MZ2] J. Ma, F. Zheng Compact hyperbolic Coxeter five-dimensional polytopes with nine facets.

[Mak] V. S. Makarov, The Fedorov groups of four-dimensional and five-dimensional Lobaĉevskiı space, from: Studies in General Algebra, No. 1 (Russian), Kiŝhinev Gos. Univ., Kiŝhinev (1968) 120–129.

[Mcl] J. Mcleod, Hyperbolic Coxeter pyramids, Advances in Pure Mathematics, 2013, 3, 78-82.

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[Rob] M. Roberts, A Classification of Non-Compact Coxeter Polytopes with n+3 Facets and One Non-Simple Vertex, arXiv:1511.08451.

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[Ruz] O. P. Ruzmanov, Examples of nonarithmetic crystallographic Coxeter groups in n–dimensional Lobachevskii space when 6≤n≤10, from: Problems in group theory and in homological algebra (Russian), Yaroslav. Gos. Univ., Yaroslavl’ (1989) 138–142.

[Tum1] P. Tumarkin, Hyperbolic Coxeter n-polytopes with n+2 facets, Math. Notes 75 (2004), 848-854, arXiv:math/0301133.

[Tum2] P. Tumarkin, Compact hyperbolic Coxeter n-polytopes with n+3 facets, Electron. J. Combin. 14 (2007), R69, arXiv:math/0406226.

[Tum3] P. Tumarkin, Hyperbolic Coxeter n-polytopes with n+3 facets, Trans. Moscow Math. Soc. (2004), 235–25.

[Tum3'] P. Tumarkin, Non-compact hyperbolic Coxeter n-polytopes with n+3 facets, short version (3 pages) Russian Math. Surveys, 58 (2003), 805-806, arXiv:math/0311272.

[Ves] A. Vesnin, Volumes and normalized volumes of right-angled hyperbolic polyhedra Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia, 57 (2010), 159-169.

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[Vin5] E. B. Vinberg, Non-arithmetic hyperbolic reflection groups in higher dimensions, Mosc. Math. J., 2015, Vol. 15, N 3, 593–602. See also preprint version.

[Vin6] E. B. Vinberg, Discrete reflection groups in Lobachevskii spaces, Proceedings of the International Congress of Mathematicians, August 16-24, 1983, Warszawa.

[Vin7] E. B. Vinberg (Ed.), Geometry II. Encyclopaedia of Mathematical Sciences, vol. 29. Springer-Verlag, Berlin, 1993. Russian version: part 1 and part 2.

This page is maintained by Anna Felikson and Pavel Tumarkin. The idea of the page is suggested by Ruth Kellerhals .