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].
- A non-compact 4-polytope with 8 facets and a unique non-simple vertex, p.11 in [CR]
- 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__

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[CR]
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[Ch]
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A. Felikson, P. Tumarkin,
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[FTZ]
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G. J. Heckman,
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[ImH1] H.-Ch. Im Hof,
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T. Inoue,
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*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,
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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,
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[Ves] A. Vesnin,
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This page is maintained by Anna Felikson and Pavel Tumarkin. The idea of the page is suggested by Ruth Kellerhals.