Hyperbolic Coxeter polytopes
Why "hyperbolic": spherical and Euclidean Coxeter polytopes are classified by H.S.M.Coxeter in 1934 [Cox].
- 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.
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).
- Products of two simplices:
- Pyramids over a product of two simplices [Tum1], dim=3,...,13, 17.
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]).
- 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].
- 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):
- 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].
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 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].
- 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].
- Compact polytopes do not exist for dim>4, examples known for dim=2,3,4
- 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
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
Volumes, subgroups, commensurability:
- Volumes of hyperbolic simplices [JKRT1].
- Volumes of compact 4-dimensional polytopes that are products of simplices [FTZ].
- Volumes in even dimensions [Heck].
- A fundamental polytope of a subgroup has more faces than the fundamental polytope of a group [FT4].
- Subgroup relations for hyperbolic simplices, [FT5].
- Commensurability of hyperbolic simplices [JKRT2], see also [FT5].
Commensurability of pyramids over a product of two simplices [GJK].
- 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
Some technical tools:
- 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.
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