# Hyperbolic Coxeter polytopes

## 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].

### 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:
• 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 [Bur]

### 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].

### 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 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].

## Software:

• CoxIter, 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.

## References:

• [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 Z[(√ 5 +1)/2], Moscow Univ. Math. Bull. 39 (1984), 6–14.
• [Bur] A. Burcroff, On compact hyperbolic Coxeter polytopes with few facetsZ[(√ 5 +1)/2], MSc Thesis, Durham (2021).
• [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.
• [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.