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].

- Definitions of Coxeter polytope, Gram matrix, Coxeter diagram.
- Faces of Coxeter polytopes.
- Existence and Uniqueness of a polytope with given Gram matrix.

- Compact hyperbolic Coxeter polytopes:
- Finite volume hyperbolic Coxeter polytopes:

- 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].

- 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.

- Products of two simplices:
- n=dim+3:
- 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]).

- p=0:
- 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].

- Pyramids:
- Cubes:
- Products of simplices: is either a cube or a product of at most 3 simplices [Alex].

- 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].

- 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:

- 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].

- Nikulin's estimate on average number of edges [Nik].
- Gale diagrams.
- Local determinants [Vin2].
- Coxeter faces of polytopes and their dihedral angles [Bor], [All].
- Missing faces and Lanner diagrams.

, applet by Rafael Guglielmetti: computes invariants of Coxeter polytopes with given Coxeter diagram. In particular, it checks cocompactness, cofiniteness, arithmeticity and computes f-vector, Euler characteristic, signature, dimension, growth series and growth rate.__CoxIter__

- [Alex] S. Alexandrov,
*Lanner diagrams and combinatorial properties of compact hyperbolic Coxeter polytopes*. - [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. - [And2] E. M. Andreev,
*On convex polyhedra of finite volume in Lobachevskii spaces,*Math. USSR Sbornik 12 (1970), 255–259, Russian version. - [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*, Moscow Univ. Math. Bull. 39 (1984), 6–14, Russian version.**Z**[(√ 5 +1)/2] - [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. - [Cox]
H. S. M. Coxeter,
*Discrete groups generated by reflections. Ann. Math. 35 (1934)*, 588--621. - [Duf] G. Dufour,
*Notes on right-angled Coxeter polyhedra in hyperbolic spaces*, Geometriae Dedicata (2010), Vol. 147, 1, 277–282. - [Ess1] F. Esselmann,
*Über kompakte, hyperbolische Coxeter-Polytope mit wenigen Facetten*. SFB 343 preprint 94-087, Bielefeld. - [Ess2] F. Esselmann,
*The classification of compact hyperbolic Coxeter d-polytopes with d + 2 facets*, Comment. Math. Helvetici 71 (1996), 229–242. - [GJK] R. Guglielmetti, M. Jacquemet, R. Kellerhals,
*On commensurable hyperbolic Coxeter groups, Geom*. Dedicata 183 (2016), 143-167. - [FT1] A. Felikson, P. Tumarkin,
*On hyperbolic Coxeter polytopes with mutually intersecting facets*, J. Combin. Theory A 115 (2008), 121-146, arXiv:math/0604248. - [FT2] A. Felikson, P. Tumarkin,
*Coxeter polytopes with a unique pair of non-intersecting facets*, J. Combin. Theory A 116 (2009), 875-902, arXiv:0706.3964. - [FT3] A. Felikson, P. Tumarkin,
*Essential hyperbolic Coxeter polytopes*, Israel Journal of Mathematics (2014), 199, 1, 113–161, arXiv:0906.4111. - [FT4] A. Felikson, P. Tumarkin,
*Reflection subgroups of Coxeter groups*, Trans. Amer. Math. Soc. 362 (2010), 847-858, arXiv:0705.0426. - [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. - [JKRT1] N. Johnson, R. Kellerhals, J. Ratcliffe, S. Tschantz,
*The size of a hyperbolic Coxeter simplex*, Transformation Groups 4 (1999), 329-353. - [JKRT2] N. Johnson, R. Kellerhals, J. Ratcliffe, S. Tschantz,
*Commensurability classes of hyperbolic Coxeter simplex reflection groups*, Linear Algebra Appl. 345 (2002), 119-147. - [Heck]
G. J. Heckman,
*The volume of hyperbolic Coxeter polytopes of even dimension,*Indag. Mathem., N.S., 6 (2), 189-196. -
[ImH1] H.-Ch. Im Hof,
*A class of hyperbolic Coxeter groups*, Expo. Math 3, 179-186 (1985). - [ImH2] H.-Ch. Im Hof,
*Napier cycles and hyperbolic Coxeter groups*, Bull. Soc. Math. de Belg. Serie A, XLII (1990), 523–545. - [In]
T. Inoue,
*Organizing volumes of right-angled hyperbolic polyhedra*, Algebraic & Geometric Topology, 8 (2008), 1523–1565. arXiv:0809.2111. - [Kap] I. M. Kaplinskaja,
*Discrete groups generated by reflections in the faces of simplicial prisms in Lobachevskian spaces*, Math. Notes 15 (1974), 88–91. Russian version. - [KV] I. M. Kaplinskaya,
E. B. Vinberg,
*On the groups O_18,1(Z) and O_19,1(Z)*, Soviet Math.Dokl. 19 (1978), 194–197. Russian version. - [Khov]
*Hyperplane sections of polyhedra, toric varieties and discrete groups in Lobachevsky space,*Functional Analysis and its applications, V. 20, N 1, 50–61, 1986; translation in Funct. Anal. Appl. V. 20 (1986), no. 1, 41–50. Russian version. - [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. - [Nik]
V. V. Nikulin,
*On the classification of arithmetic groups generated by reflections in Lobachevsky spaces*, Math. USSR Izv. 18 (1982), 99–123. Russian version. - [Non]
J. Nonaka,
*The number of cusps of right-angled polyhedra in hyperbolic spaces*, arXiv:1312.0380. - [Pog] Pogorelov,
*A regular partition of Lobachevskian space*Mathematical Notes of the Academy of Sciences of the USSR (1967) 1:3. Russian version. - [Po] H. Poincaré,
*Théorie des groups fuchsiennes*, Acta Math. 1 (1882), 1--62. - [PV] L. Potyagailo, E. Vinberg,
*On right-angled reflection groups in hyperbolic spaces*, Comment. Math. Helv. 80 (2005), 63–73. - [Pr1] M. N. Prokhorov,
*The absence of discrete reflection groups with non-compact fundamental polyhedron of finite volume in Lobachevskij spaces of large dimension*, Math. USSR Izv. 28 (1987), 401–411. Russian version. - [Pr2]
M. N. Prokhorov,
*On polyhedra of finite volume in the Lobachevskii space with dihedral angles π/2 and π/3*(Russian), Lectures in mathematics and its applications, Vol. 2, No. 2 (Russian), 151–187, Ross. Akad. Nauk, Inst. Mat. im. Steklova, Moscow, 1988. - [Rob] M. Roberts,
*A Classification of Non-Compact Coxeter Polytopes with n+3 Facets and One Non-Simple Vertex*, arXiv:1511.08451. - [RHD]
R.K.W. Roeder, J. H. Hubbard, W. D. Dunbar,
*Andreev’s Theorem on hyperbolic polyhedra*, Annales de l’institut Fourier (2007) Vol. 57, Issue 3, 825-882. arXiv:math/0601146. - [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. - [Vin1] E. B. Vinberg,
*Discrete groups generated by reflections in Lobachevskii spaces*, Mat. USSR sb. 1 (1967), 429–444. Russian version. - [Vin2] E. B. Vinberg,
*The absence of crystallographic groups of reflections in Lobachevsky spaces of large dimensions*, Trans. Moscow Math. Soc. 47 (1985), 75–112. Russian version. - [Vin3] E. B. Vinberg,
*Hyperbolic reflection groups*, Russian Math. Surveys 40 (1985), 31–75. Russian version. - [Vin4] E. B. Vinberg,
*On groups of unit elements of certain quadratic forms*, Math. USSR Sb. 16 (1972), 17–35. Russian version. - [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.