Hyperbolic
reflection
groups
• Reflection is a symmetry with respect to a hyperplane.
• Reflection is a symmetry with respect to a hyperplane.
• Reflection group is a group generated by reflections.
• Reflection is a symmetry with respect to a hyperplane.
• Reflection group is a group generated by reflections.
Example: dihedral group D2n
(symmetry group of n-gon)
• Reflection is a symmetry with respect to a hyperplane.
• Reflection group is a group generated by reflections.
Example: dihedral group D2n
(symmetry group of n-gon)
• generated by reflections ra and rb
a
b
• Reflection is a symmetry with respect to a hyperplane.
• Reflection group is a group generated by reflections.
Example: dihedral group D2n
(symmetry group of n-gon)
• generated by reflections ra and rb
• (rarb)n = e
a
b
• Reflection is a symmetry with respect to a hyperplane.
• Reflection group is a group generated by reflections.
Example: dihedral group D2n
(symmetry group of n-gon)
• generated by reflections ra and rb
• (rarb)n = e
• D2n = hra, rb | ra2 = rb2 = (rarb)n = ei
a
b
X = Ed, Sd or Hd, group G : X.
X = Ed, Sd or Hd, group G : X.
G is discrete if for any x, y ∈ X there exists a neighbourhood Ox
containing only finitely many points of the orbit γy, γ ∈ G.
X = Ed, Sd or Hd, group G : X.
G is discrete if for any x, y ∈ X there exists a neighbourhood Ox
containing only finitely many points of the orbit γy, γ ∈ G.
Example: G = hra, rbi.
b
a
X = Ed, Sd or Hd, group G : X.
G is discrete if for any x, y ∈ X there exists a neighbourhood Ox
containing only finitely many points of the orbit γy, γ ∈ G.
Example: G = hra, rbi.
b
a
G is discrete ⇔ ∠ab =
kπ
l ,
k, l ∈ Z.
X = Ed, Sd or Hd, group G : X.
G is discrete if for any x, y ∈ X there exists a neighbourhood Ox
containing only finitely many points of the orbit γy, γ ∈ G.
Example: G = hra, rbi.
b
a
G is discrete ⇔ ∠ab =
kπ
l ,
k, l ∈ Z.
X = Ed, Sd or Hd, group G : X.
G is discrete if for any x, y ∈ X there exists a neighbourhood Ox
containing only finitely many points of the orbit γy, γ ∈ G.
Example: G = hra, rbi.
b
a
G is discrete ⇔ ∠ab =
kπ
l ,
k, l ∈ Z.
X = Ed, Sd or Hd, group G : X.
G is discrete if for any x, y ∈ X there exists a neighbourhood Ox
containing only finitely many points of the orbit γy, γ ∈ G.
Example: G = hra, rbi.
b
a
G is discrete ⇔ ∠ab =
kπ
l ,
k, l ∈ Z.
X = Ed, Sd or Hd, group G : X.
G is discrete if for any x, y ∈ X there exists a neighbourhood Ox
containing only finitely many points of the orbit γy, γ ∈ G.
Example: G = hra, rbi.
b
a
G is discrete ⇔ ∠ab =
kπ
l ,
k, l ∈ Z.
X = Ed, Sd or Hd, group G : X.
G is discrete if for any x, y ∈ X there exists a neighbourhood Ox
containing only finitely many points of the orbit γy, γ ∈ G.
Example: G = hra, rbi.
b
a
G is discrete ⇔ ∠ab =
kπ
l ,
k, l ∈ Z.
X = Ed, Sd or Hd, group G : X.
G is discrete if for any x, y ∈ X there exists a neighbourhood Ox
containing only finitely many points of the orbit γy, γ ∈ G.
Example: G = hra, rbi.
b
a
G is discrete ⇔ ∠ab =
kπ
l ,
k, l ∈ Z.
X = Ed, Sd or Hd, group G : X.
G is discrete if for any x, y ∈ X there exists a neighbourhood Ox
containing only finitely many points of the orbit γy, γ ∈ G.
Example: G = hra, rbi.
b
a
G is discrete ⇔ ∠ab =
kπ
l ,
k, l ∈ Z.
X = Ed, Sd or Hd, group G : X.
G is discrete if for any x, y ∈ X there exists a neighbourhood Ox
containing only finitely many points of the orbit γy, γ ∈ G.
Example: G = hra, rbi.
b
a
G is discrete ⇔ ∠ab =
kπ
l ,
k, l ∈ Z.
X = Ed, Sd or Hd, group G : X.
G is discrete if for any x, y ∈ X there exists a neighbourhood Ox
containing only finitely many points of the orbit γy, γ ∈ G.
Example: G = hra, rbi.
G is discrete ⇔ ∠ab =
• Any finite group is discrete.
kπ
l ,
k, l ∈ Z.
X = Ed, Sd or Hd, group G : X.
G is discrete if for any x, y ∈ X there exists a neighbourhood Ox
containing only finitely many points of the orbit γy, γ ∈ G.
Example: G = hra, rbi.
G is discrete ⇔ ∠ab =
kπ
l ,
• Any finite group is discrete.
• Any discrete group stabilizing some point is finite.
k, l ∈ Z.
X = Ed, Sd or Hd, group G : X.
G is discrete if for any x, y ∈ X there exists a neighbourhood Ox
containing only finitely many points of the orbit γy, γ ∈ G.
Example: G = hra, rbi.
G is discrete ⇔ ∠ab =
kπ
l ,
k, l ∈ Z.
• Any finite group is discrete.
• Any discrete group stabilizing some point is finite.
A set F ⊂ X is a fundamental domain of a discrete group G if
S
X = γ∈G γF and F ∩ γF ⊂ ∂F for all γ ∈ G.
X = Ed, Sd or Hd, group G : X.
G is discrete if for any x, y ∈ X there exists a neighbourhood Ox
containing only finitely many points of the orbit γy, γ ∈ G.
Example: G = hra, rbi.
G is discrete ⇔ ∠ab =
kπ
l ,
k, l ∈ Z.
• Any finite group is discrete.
• Any discrete group stabilizing some point is finite.
A set F ⊂ X is a fundamental domain of a discrete group G if
S
X = γ∈G γF and F ∩ γF ⊂ ∂F for all γ ∈ G.
Example: G = hra, rbi.
∠ab is a fundamental domain of G ⇔ ∠ab = πl , l ∈ Z.
Example: Let ∠ab =
2π
3 .
Example: Let ∠ab =
2π
3 .
Example: Let ∠ab =
2π
3 .
Example: Let ∠ab =
2π
3 .
Example: Let ∠ab =
2π
3 .
Example: Let ∠ab =
2π
3 .
Example: regular tetrahedron T
Example: regular tetrahedron T
group of symmetries Sym(T )?
Example: regular tetrahedron T
group of symmetries Sym(T )?
Example: regular tetrahedron T
group of symmetries Sym(T )?
Example: regular tetrahedron T
group of symmetries Sym(T )?
• Sym(T ) is generated by 3 reflections.
Example: regular tetrahedron T
group of symmetries Sym(T )?
• Sym(T ) is generated by 3 reflections.
• All mirrors cut T into 24 tetrahedra
Example: regular tetrahedron T
group of symmetries Sym(T )?
• Sym(T ) is generated by 3 reflections.
• All mirrors cut T into 24 tetrahedra
(fundamental domains of Sym(T )).
Example: regular tetrahedron T
group of symmetries Sym(T )?
• Sym(T ) is generated by 3 reflections.
• All mirrors cut T into 24 tetrahedra
(fundamental domains of Sym(T )).
Example: regular tetrahedron T
group of symmetries Sym(T )?
• Sym(T ) is generated by 3 reflections.
• All mirrors cut T into 24 tetrahedra
6
(fundamental domains of Sym(T )).
Example: regular tetrahedron T
group of symmetries Sym(T )?
• Sym(T ) is generated by 3 reflections.
• All mirrors cut T into 24 tetrahedra
6 4
(fundamental domains of Sym(T )).
Example: regular tetrahedron T
group of symmetries Sym(T )?
• Sym(T ) is generated by 3 reflections.
6
6 4
• All mirrors cut T into 24 tetrahedra
(fundamental domains of Sym(T )).
Example: regular tetrahedron T
group of symmetries Sym(T )?
• Sym(T ) is generated by 3 reflections.
6
6 4
• All mirrors cut T into 24 tetrahedra
(fundamental domains of Sym(T )).
Example: regular tetrahedron T
group of symmetries Sym(T )?
• Sym(T ) is generated by 3 reflections.
6
6 4
• All mirrors cut T into 24 tetrahedra
(fundamental domains of Sym(T )).
• S2 is tiled by 24 triangles with angles ( π3 , π2 , π3 )
Example: regular tetrahedron T
group of symmetries Sym(T )?
• Sym(T ) is generated by 3 reflections.
6
6 4
• All mirrors cut T into 24 tetrahedra
(fundamental domains of Sym(T )).
• S2 is tiled by 24 triangles with angles ( π3 , π2 , π3 )
• Sym(T ) = hr1, r2, r3 | r12 = r22 = r32 = e,
(r1r3)2 = (r1r2)3 = (r2r3)3 = ei
Example: regular tetrahedron T
group of symmetries Sym(T )?
• Sym(T ) is generated by 3 reflections.
6
6 4
• All mirrors cut T into 24 tetrahedra
(fundamental domains of Sym(T )).
• S2 is tiled by 24 triangles with angles ( π3 , π2 , π3 )
• Sym(T ) = hr1, r2, r3 | r12 = r22 = r32 = e,
(r1r3)2 = (r1r2)3 = (r2r3)3 = ei
• Sym(T ) = S4
Example: regular tetrahedron T
group of symmetries Sym(T )?
• Sym(T ) is generated by 3 reflections.
6
6 4
• All mirrors cut T into 24 tetrahedra
(fundamental domains of Sym(T )).
• S2 is tiled by 24 triangles with angles ( π3 , π2 , π3 )
• Sym(T ) = hr1, r2, r3 | r12 = r22 = r32 = e,
(r1r3)2 = (r1r2)3 = (r2r3)3 = ei
• Sym(T ) = S4,
Sym(T ) permutes faces of T
Example: regular tetrahedron T
group of symmetries Sym(T )?
• Sym(T ) is generated by 3 reflections.
6
6 4
• All mirrors cut T into 24 tetrahedra
(fundamental domains of Sym(T )).
• S2 is tiled by 24 triangles with angles ( π3 , π2 , π3 )
• Sym(T ) = hr1, r2, r3 | r12 = r22 = r32 = e,
(r1r3)2 = (r1r2)3 = (r2r3)3 = ei
• Sym(T ) = S4,
Sym(T ) permutes faces of T
• S4 is gen. by transpositions ↔ Sym(T ) is gen. by reflections
Example: regular 3-polytopes
Example: regular 3-polytopes
• Sym(P ) is generated by 3 symmetries
Example: regular 3-polytopes
( π3 , π2 , π4 )
( π3 , π2 , π5 )
( π3 , π2 , π3 )
( π4 , π2 , π3 )
• Sym(P ) is generated by 3 symmetries
( π5 , π2 , π3 )
Regular d-polytopes
• all facets (codim. 1 faces) are congruent regular polytopes.
• all dihedral angles are equal.
Regular d-polytopes
• all facets (codim. 1 faces) are congruent regular polytopes.
• all dihedral angles are equal.
Regular d-polytopes
• all facets (codim. 1 faces) are congruent regular polytopes.
• all dihedral angles are equal.
Regular d-polytopes
• all facets (codim. 1 faces) are congruent regular polytopes.
• all dihedral angles are equal.
Regular d-polytopes
• all facets (codim. 1 faces) are congruent regular polytopes.
• all dihedral angles are equal.
• Sym(P ) is generated by d reflections.
In general: if G : X is a discrete reflection group
In general: if G : X is a discrete reflection group
• X is decomposed by hyperplanes of reflections (mirrors)
into chambers.
In general: if G : X is a discrete reflection group
• X is decomposed by hyperplanes of reflections (mirrors)
into chambers.
• Chambers are congruent.
In general: if G : X is a discrete reflection group
• X is decomposed by hyperplanes of reflections (mirrors)
into chambers.
• Chambers are congruent.
• Chambers are fundamental domains of G
(chambers ↔ elements of G)
In general: if G : X is a discrete reflection group
• X is decomposed by hyperplanes of reflections (mirrors)
into chambers.
• Chambers are congruent.
• Chambers are fundamental domains of G
(chambers ↔ elements of G)
• Each chamber is a convex polytope
(i.e. intersection of countable number of halfspaces).
In general: if G : X is a discrete reflection group
• X is decomposed by hyperplanes of reflections (mirrors)
into chambers.
• Chambers are congruent.
• Chambers are fundamental domains of G
(chambers ↔ elements of G)
• Each chamber is a convex polytope
(i.e. intersection of countable number of halfspaces).
• Chambers have good dihedral angles (otherwise G is not discrete).
• A polytope P is called a Coxeter polytope
if all dihedral angles of P are submultiples of π.
• A polytope P is called a Coxeter polytope
if all dihedral angles of P are integer submultiples of π.
•
Discrete reflection
group G
−→
Coxeter polytope
(chamber of G)
• A polytope P is called a Coxeter polytope
if all dihedral angles of P are integer submultiples of π.
•
Discrete reflection
group G
• Coxeter polytope P
−→
Coxeter polytope
(chamber of G)
−→
Group GP gen. by reflections
across facets of P
• A polytope P is called a Coxeter polytope
if all dihedral angles of P are integer submultiples of π.
•
Discrete reflection
group G
• Coxeter polytope P
−→
Coxeter polytope
(chamber of G)
−→
Group GP gen. by reflections
across facets of P
◦ GP is discrete and P is its fundamental domain.
• A polytope P is called a Coxeter polytope
if all dihedral angles of P are integer submultiples of π.
•
Discrete reflection
group G
• Coxeter polytope P
−→
Coxeter polytope
(chamber of G)
−→
Group GP gen. by reflections
across facets of P
◦ GP is discrete and P is its fundamental domain.
◦ GP = hrfi , fi ∈ {facets of P } | rf2i = e, (rfi rfj )mij = ei
where ∠fifj =
π
mij
if fi ∩ fj 6= ∅ and mij = ∞ otherwise.
• A polytope P is called a Coxeter polytope
if all dihedral angles of P are integer submultiples of π.
•
Discrete reflection
group G
• Coxeter polytope P
−→
Coxeter polytope
(chamber of G)
−→
Group GP gen. by reflections
across facets of P
◦ GP is discrete and P is its fundamental domain.
◦ GP = hrfi , fi ∈ {facets of P } | rf2i = e, (rfi rfj )mij = ei
where ∠fifj =
π
mij
i.e. GP is a Coxeter group.
if fi ∩ fj 6= ∅ and mij = ∞ otherwise.
Examples of infinite groups:
• we are mainly interested in finite volume groups,
where vol(X/G) < ∞.
How to describe a reflection?
v
x
How to describe a reflection?
v
xv
x
How to describe a reflection?
v
xv
x
Rv (x)
Rv (x) = x − 2xv
How to describe a reflection?
v
xv
x
xv = |x| cos α · √ v
(v,v)
Rv (x)
Rv (x) = x − 2xv
How to describe a reflection?
v
xv
x
xv = |x| cos α · √ v
(v,v)
=
Rv (x)
Rv (x) = x − 2xv
p
(x,v)
(x, x) √
(x,x)(v,v)
·√v
(v,v)
How to describe a reflection?
v
xv
x
xv = |x| cos α · √ v
(v,v)
Rv (x)
Rv (x) = x − 2xv
(x,v)
(x, x) √
=
p
=
(x,v)
(v,v) v
(x,x)(v,v)
·√v
(v,v)
How to describe a reflection?
v
xv
x
xv = |x| cos α · √ v
(v,v)
Rv (x)
Rv (x) = x − 2xv
= x − 2 (x,v)
(v,v) v
(x,v)
(x, x) √
=
p
=
(x,v)
(v,v) v
(x,x)(v,v)
·√v
(v,v)
How to describe a reflection?
Sd ⊂ Rd+1
v
xv
x
xv = |x| cos α · √ v
(v,v)
Rv (x)
Rv (x) = x − 2xv
= x − 2 (x,v)
(v,v) v
(x,v)
(x, x) √
=
p
=
(x,v)
(v,v) v
(x,x)(v,v)
·√v
(v,v)
How to describe a reflection?
Sd ⊂ Rd+1
v
xv
x
Ed
v
x
a
Rv (x)
Rv (x) = x − 2xv
= x − 2 (x,v)
(v,v) v
How to describe a reflection?
Sd ⊂ Rd+1
v
xv
x
Ed
v
x−a
x
a
Rv (x)
Rv (x) = x − 2xv
= x − 2 (x,v)
(v,v) v
How to describe a reflection?
Sd ⊂ Rd+1
v
xv
x
Ed
v
x−a
x
a
Rv (x)
Rv (x) = x − 2xv
= x − 2 (x,v)
(v,v) v
Rv (x − a)
How to describe a reflection?
Sd ⊂ Rd+1
v
xv
x
Ed
v
x−a
x
a
Rv (x)
Rv (x) = x − 2xv
= x − 2 (x,v)
(v,v) v
Rv (x − a)
Rv (x − a) + a
Linear model of Hd
Rd,1:
(u, v) = −u0v0 + u1v1 + u2v2 + · · · + udvd
Linear model of Hd
Rd,1:
(u, v) = −u0v0 + u1v1 + u2v2 + · · · + udvd
• Points of Hd
↔ {u ∈ Rd,1 | u2 = −1, u0 > 0}.
x0
Linear model of Hd
Rd,1:
(u, v) = −u0v0 + u1v1 + u2v2 + · · · + udvd
• Points of Hd
↔ {u ∈ Rd,1 | u2 = −1, u0 > 0}.
• Points of ∂Hd
↔ {u ∈ Rd,1 | u2 = 0, u0 > 0}.
x0
Linear model of Hd
Rd,1:
(u, v) = −u0v0 + u1v1 + u2v2 + · · · + udvd
• Points of Hd
↔ {u ∈ Rd,1 | u2 = −1, u0 > 0}.
• Points of ∂Hd
↔ {u ∈ Rd,1 | u2 = 0, u0 > 0}.
• Hyperplane Hu ↔ {u ∈ Rd,1 | u2 = 1}.
x0
Linear model of Hd
Rd,1:
x0
(u, v) = −u0v0 + u1v1 + u2v2 + · · · + udvd
• Points of Hd
↔ {u ∈ Rd,1 | u2 = −1, u0 > 0}.
• Points of ∂Hd
↔ {u ∈ Rd,1 | u2 = 0, u0 > 0}.
• Hyperplane Hu ↔ {u ∈ Rd,1 | u2 = 1}.
• Pair of hyperplanes: (u, v) = − cos(∠(Hu, Hv )) ↔ Hu ∩ Hv 6= ∅
(u, v) = −1
↔ Hu ∩ Hv ⊂ ∂Hd
(u, v) = − cosh ρ(Hu, Hv ) ↔ Hu ∩ Hv = ∅
Linear model of Hd
Rd,1:
x0
(u, v) = −u0v0 + u1v1 + u2v2 + · · · + udvd
• Points of Hd
↔ {u ∈ Rd,1 | u2 = −1, u0 > 0}.
• Points of ∂Hd
↔ {u ∈ Rd,1 | u2 = 0, u0 > 0}.
• Hyperplane Hu ↔ {u ∈ Rd,1 | u2 = 1}.
• Pair of hyperplanes: (u, v) = − cos(∠(Hu, Hv )) ↔ Hu ∩ Hv 6= ∅
(u, v) = −1
↔ Hu ∩ Hv ⊂ ∂Hd
(u, v) = − cosh ρ(Hu, Hv ) ↔ Hu ∩ Hv = ∅
• Reflection across Hv :
Rv (x) = x − 2 (x,v)
(v,v) v.
Gram matrix
P ⊂ Sd, Ed or Hd
• gii = 1,
−→ Symmetric matrix

π

−
cos(

mij ),

gij = −1,


− cosh(ρ(f , f )),
i
j
G(P ) = {gij }
if ∠(fifj ) = π/mij ,
if fi is parallel to fj ,
if fi and fj diverge.
Gram matrix
P ⊂ Sd, Ed or Hd
• gii = 1,
−→ Symmetric matrix

π

−
cos(

mij ),

gij = −1,


− cosh(ρ(f , f )),
i
j
G(P ) = {gij }
if ∠(fifj ) = π/mij ,
if fi is parallel to fj ,
if fi and fj diverge.
Gram matrix
P ⊂ Sd, Ed or Hd
• gii = 1,
−→ Symmetric matrix

π

−
cos(

mij ),

gij = −1,


− cosh(ρ(f , f )),
i
Sd
j
G(P ) = {gij }
if ∠(fifj ) = π/mij ,
if fi is parallel to fj ,
if fi and fj diverge.
Gram matrix
P ⊂ Sd, Ed or Hd
• gii = 1,
−→ Symmetric matrix

π

−
cos(

mij ),

gij = −1,


− cosh(ρ(f , f )),
i
Sd
Ed
j
G(P ) = {gij }
if ∠(fifj ) = π/mij ,
if fi is parallel to fj ,
if fi and fj diverge.
Gram matrix
P ⊂ Sd, Ed or Hd
• gii = 1,
−→ Symmetric matrix

π

−
cos(

mij ),

gij = −1,


− cosh(ρ(f , f )),
i
Sd
Ed
j
G(P ) = {gij }
if ∠(fifj ) = π/mij ,
if fi is parallel to fj ,
if fi and fj diverge.
Hd
Gram matrix
P ⊂ Sd, Ed or Hd
• gii = 1,
−→ Symmetric matrix

π

−
cos(

mij ),

gij = −1,


− cosh(ρ(f , f )),
i
Sd
sgn (G(P )) :
Ed
(d + 1, 0)
j
G(P ) = {gij }
if ∠(fifj ) = π/mij ,
if fi is parallel to fj ,
if fi and fj diverge.
Hd
(d, 0, 1)
(d, 1)
Coxeter diagram Σ(P )
• Nodes ←→ facets fi of P
• Edges:
if ∠(fifj ) = π/2
mij
if ∠(fifj ) = π/mij
if ∠(fifj ) = π/3
if ∠(fifj ) = π/4
if ∠(fifj ) = π/5
if fi ∩ fj = ∅
if fi ∩ fj ∈ ∂Hd
Examples:
Coxeter diagram Σ(P )
• Nodes ←→ facets fi of P
• Edges:
Examples:
if ∠(fifj ) = π/2
mij
if ∠(fifj ) = π/mij
if ∠(fifj ) = π/3
if ∠(fifj ) = π/4
if ∠(fifj ) = π/5
if fi ∩ fj = ∅
if fi ∩ fj ∈ ∂Hd
6
Coxeter polytopes in Sd, Ed and Hd:
• P ⊂ Sd. Finitely many in each dimension,
Classified (Coxeter, 1934).
• P ⊂ Ed. Finitely many in each dimension,
Classified (Coxeter, 1934).
• P ⊂ Hd. Infinitely many, No classification.
Spherical Coxeter polytopes
• P ⊂ Sd ⇒ P is a simplex.
• G(P ) > 0
◦ no
⇒ Any connected component of Σ(P ) has
k
for k > 5,
◦ no cycles,
◦ at most one multiple edge,
◦ no nodes of valency ≥ 4,
◦ at most one node of valency 3.
Spherical Coxeter polytopes
• P ⊂ Sd ⇒ P is a simplex.
• Coxeter diagram of P is called elliptic, it is a union of
(m)
G2
m
E6
dihedron
E7
An
simplex
E8
Bn = Cn
Dn
cube, cocube
F4
H3
H4
24-cell
dodecahedron,
icosohedron
120-cell
Regular polytopes: classification
• P ⊂ Sd ⇒ P is a simplex.
• Coxeter diagram of P is called elliptic, it is a union of
(m)
G2
m
E6
×
E7
×
E8
×
dihedron
An
simplex
Bn = Cn
cube, cocube
Dn
×
F4
H3
H4
24-cell
dodecahedron,
icosohedron
120-cell,
600-cell
Regular polytopes: classification
• Regular polytopes correspond to linear elliptic diagrams:
• Coxeter diagram of P is called elliptic, it is a union of
(m)
G2
m
E6
×
E7
×
E8
×
dihedron
An
simplex
Bn = Cn
cube, cocube
Dn
×
F4
H3
H4
24-cell
dodecahedron,
icosohedron
120-cell,
600-cell
Euclidean Coxeter polytopes
• P ⊂ Ed ⇒ P is a product of simplices.
• Coxeter diagram of P is called parabolic, it is a union of
Euclidean Coxeter polytopes
• P ⊂ Ed ⇒ P is a product of simplices.
e1
A
1
0
0
1
0
1
0
1
0
1
0
1
• Coxeter diagram of P is called parabolic, it is a union of
e6
E
en
A
e7
E
en
B
en
C
e8
E
Fe4
en
D
e2
G
Acute-angled polytopes in Ed, Sd and Hd
A polytope is acute-angled if all its dihedral angles are acute
(α is called acute if α ≤ π/2).
Thm. A face of an acute-angled polytope is acute-angled.
• Prove the theorem for a facet and use induction.
+
+ T
k
• A polytope P = ∩Hi , a facet f = ∩i=0Hi
H.
• Angles of H are
∠(H ∩ Hi, H ∩ Hj ).
• Consider a section by a 3-plane
Π3 ⊥ (H ∩ Hi ∩ Hj ):
•a≤α
Acute-angled polytopes in Ed, Sd and Hd
A polytope is acute-angled if all its dihedral angles are acute
(α is called acute if α ≤ π/2).
Thm. A face of an acute-angled polytope is acute-angled.
• Prove the theorem for a facet and use induction.
+
+ T
k
• A polytope P = ∩Hi , a facet f = ∩i=0Hi
H.
• Angles of H are
∠(H ∩ Hi, H ∩ Hj ).
• Consider a section by a 3-plane
Π3 ⊥ (H ∩ Hi ∩ Hj ):
•a≤α
Acute-angled polytopes in Ed, Sd and Hd
A polytope is acute-angled if all its dihedral angles are acute
(α is called acute if α ≤ π/2).
Thm. A face of an acute-angled polytope is acute-angled.
• Prove the theorem for a facet and use induction.
+
+ T
k
• A polytope P = ∩Hi , a facet f = ∩i=0Hi
H.
• Angles of H are
∠(H ∩ Hi, H ∩ Hj ).
• Consider a section by a 3-plane
Π3 ⊥ (H ∩ Hi ∩ Hj ):
•a≤α
Acute-angled polytopes in Ed, Sd and Hd
A polytope is acute-angled if all its dihedral angles are acute
(α is called acute if α ≤ π/2).
Thm. A face of an acute-angled polytope is acute-angled.
• Prove the theorem for a facet and use induction.
+
+ T
k
• A polytope P = ∩Hi , a facet f = ∩i=0Hi
H.
• Angles of H are
∠(H ∩ Hi, H ∩ Hj ).
• Consider a section by a 3-plane
Π3 ⊥ (H ∩ Hi ∩ Hj ):
•a≤α
Acute-angled polytopes in Ed, Sd and Hd
A polytope is acute-angled if all its dihedral angles are acute
(α is called acute if α ≤ π/2).
Thm. A face of an acute-angled polytope is acute-angled.
• Prove the theorem for a facet and use induction.
+
+ T
k
• A polytope P = ∩Hi , a facet f = ∩i=0Hi
H.
• Angles of f are
∠(H ∩ Hi, H ∩ Hj ).
• Consider a section by a 3-plane
Π3 ⊥ (H ∩ Hi ∩ Hj ):
•a≤α
Acute-angled polytopes in Ed, Sd and Hd
A polytope is acute-angled if all its dihedral angles are acute
(α is called acute if α ≤ π/2).
Thm. A face of an acute-angled polytope is acute-angled.
• Prove the theorem for a facet and use induction.
+
+ T
k
• A polytope P = ∩Hi , a facet f = ∩i=0Hi
H.
• Angles of f are
∠(H ∩ Hi, H ∩ Hj ).
• Consider a section by a 3-plane
Π3 ⊥ (H ∩ Hi ∩ Hj ):
•a≤α
Π3 ∩H
a
Π3 ∩Hi
Π3 ∩Hj
α
Acute-angled polytopes in Ed, Sd and Hd
A polytope is acute-angled if all its dihedral angles are acute
(α is called acute if α ≤ π/2).
Thm. A face of an acute-angled polytope is acute-angled.
• Prove the theorem for a facet and use induction.
+
+ T
k
• A polytope P = ∩Hi , a facet f = ∩i=0Hi
H.
• Angles of f are
∠(H ∩ Hi, H ∩ Hj ).
• Consider a section by a 3-plane
Π3 ⊥ (H ∩ Hi ∩ Hj ):
•a≤α
Π3 ∩H
a
Π3 ∩Hi
Π3 ∩Hj
α
Thm. Any acute-angled polytope P ⊂ Sd containing no opposite
points of Sd is a simplex.
P
• If d = 2:
αi ≥ (n − 2)π ⇒ n = 3.
• If d > 2:
Induction on d.
◦ By ind. assumption facets are simplices, i.e. P is simplicial.
◦ By ind. assumption P is also simple
(i.e. each i-face belongs to exactly d − i facets).
◦ Simple and simplicial ⇒ simplex.
Cor. Any compact polytope in Ed and Hd is simple.
Thm. Any acute-angled polytope P ⊂ Sd containing no opposite
points of Sd is a simplex.
P
• If d = 2:
αi > (n − 2)π ⇒ n = 3.
• If d > 2:
Induction on d.
◦ By ind. assumption facets are simplices, i.e. P is simplicial.
◦ By ind. assumption P is also simple
(i.e. each i-face belongs to exactly d − i facets).
◦ Simple and simplicial ⇒ simplex.
Cor. Any compact polytope in Ed and Hd is simple.
Thm. Any acute-angled polytope P ⊂ Sd containing no opposite
points of Sd is a simplex.
P
• If d = 2:
αi > (n − 2)π ⇒ n = 3.
• If d > 2:
Induction on d.
◦ By ind. assumption facets are simplices, i.e. P is simplicial.
◦ By ind. assumption P is also simple
(i.e. each i-face belongs to exactly d − i facets).
◦ Simple and simplicial ⇒ simplex.
Cor. Any compact polytope in Ed and Hd is simple.
Thm. Any acute-angled polytope P ⊂ Sd containing no opposite
points of Sd is a simplex.
P
• If d = 2:
αi > (n − 2)π ⇒ n = 3.
• If d > 2:
Induction on d.
◦ By ind. assumption facets are simplices, i.e. P is simplicial.
◦ By ind. assumption P is also simple
(i.e. each i-face belongs to exactly d − i facets).
◦ Simple and simplicial ⇒ simplex.
Cor. Any compact polytope in Ed and Hd is simple.
Thm. Any acute-angled polytope P ⊂ Sd containing no opposite
points of Sd is a simplex.
P
• If d = 2:
αi > (n − 2)π ⇒ n = 3.
• If d > 2:
Induction on d.
◦ By ind. assumption facets are simplices, i.e. P is simplicial.
◦ By ind. assumption P is also simple
(i.e. each i-face belongs to exactly d − i facets).
◦ Simple and simplicial ⇒ simplex.
Cor. Any compact polytope in Ed and Hd is simple.
Thm. Any acute-angled polytope P ⊂ Sd containing no opposite
points of Sd is a simplex.
P
• If d = 2:
αi > (n − 2)π ⇒ n = 3.
• If d > 2:
Induction on d.
◦ By ind. assumption facets are simplices, i.e. P is simplicial.
◦ By ind. assumption P is also simple
(i.e. each i-face belongs to exactly d − i facets).
◦ Simple and simplicial ⇒ simplex.
Cor. Any compact polytope in Ed and Hd is simple.
Thm. Any acute-angled polytope P ⊂ Sd containing no opposite
points of Sd is a simplex.
P
• If d = 2:
αi > (n − 2)π ⇒ n = 3.
• If d > 2:
Induction on d.
◦ By ind. assumption facets are simplices, i.e. P is simplicial.
◦ By ind. assumption P is also simple
(i.e. each i-face belongs to exactly d − i facets).
◦ Simple and simplicial ⇒ simplex.
Cor. Any compact Coxeter polytope in Ed and Hd is simple.
Thm. Any acute-angled polytope in Ed is a direct product of several
simplices and a simplicial cone.
Lemma. L = {e1, ..., es} indecomposable system of vectors in Ed,
(ei, ej ) ≤ 0, i 6= j. Then L is either linearly independent or there
is a unique linear dependence with positive coefficients.
Hyperbolic
reflection
groups
Spherical Coxeter polytopes
• P ⊂ Sd ⇒ P is a simplex.
• Coxeter diagram of P is called elliptic, it is a union of
Gm
2
m
E6
dihedron
E7
An
simplex
E8
Bn = Cn
Dn
cube, cocube
F4
H3
H4
24-cell
dodecahedron,
icosohedron
120-cell
Euclidean Coxeter polytopes
• P ⊂ Ed ⇒ P is a product of simplices.
e1
A
1
0
0
1
0
1
0
1
0
1
0
1
• Coxeter diagram of P is called parabolic, it is a union of
e6
E
en
A
e7
E
en
B
en
C
e8
E
Fe4
en
D
e2
G
Hyperbolic Coxeter polytopes
• Variety of compact and finite-volume polytopes.
Hyperbolic Coxeter polytopes
• Variety of compact and finite-volume polytopes.
Example: Right-angled pentagon
Hyperbolic Coxeter polytopes
• Variety of compact and finite-volume polytopes.
Example: Right-angled pentagon
Hyperbolic Coxeter polytopes
• Variety of compact and finite-volume polytopes.
Example: Right-angled pentagon
Hyperbolic Coxeter polytopes
• Variety of compact and finite-volume polytopes.
Example: Right-angled pentagon
Hyperbolic Coxeter polytopes
• Variety of compact and finite-volume polytopes.
Example: Right-angled pentagon
Hyperbolic Coxeter polytopes
• Variety of compact and finite-volume polytopes.
Example: Right-angled pentagon
Hyperbolic Coxeter polytopes
• Variety of compact and finite-volume polytopes.
Example: Right-angled pentagon
Hyperbolic Coxeter polytopes
• Variety of compact and finite-volume polytopes.
− Any number of facets
− Any complexity of combinatorial types
− Arbitrary small dihedral angles
Hyperbolic Coxeter polytopes
• Variety of compact and finite-volume polytopes.
− Any number of facets
− Any complexity of combinatorial types
− Arbitrary small dihedral angles
• Thm. (Allcock’ 05) There are infinitely many finite-volume
Coxeter polytopes in Hd, for every d ≤ 19.
There are infinitely many compact Coxeter polytopes in Hd,
for every d ≤ 6.
• If P is compact then P is simple.
(i.e. d facets through each vertex)
• If P is compact then P is simple.
(i.e. d facets through each vertex)
• If P is compact then P is simple.
• If P is compact (finite volume) then P is indecomposable.
“decomposable” means
G(P ) =
• If P is compact then P is simple.
• If P is compact (finite volume) then P is indecomposable.
“decomposable” means
G(P ) =
sgn = (k, 1)
• If P is compact then P is simple.
• If P is compact (finite volume) then P is indecomposable.
“decomposable” means
G(P ) =
a subgroup H
sgn = (k, 1)
• If P is compact then P is simple.
• If P is compact (finite volume) then P is indecomposable.
“decomposable” means
G(P ) =
sgn = (k, 1)
a subgroup H, vol(FH ) = ∞
• If P is compact then P is simple.
• If P is compact (finite volume) then P is indecomposable.
“decomposable” means
G(P ) =
sgn = (k, 1)
a subgroup H, vol(FH ) = ∞
[G : H] < ∞
• If P is compact then P is simple.
• If P is compact (finite volume) then P is indecomposable.
“decomposable” means
G(P ) =
sgn = (k, 1)
a subgroup H, vol(FH ) = ∞
[G : H] < ∞,
S
FH =
gF
g∈H
G = H × K, |K| < ∞
P ,aasdasdasdqweqw
Σ(P )
− k-face f
P ,aasdasdasdqweqw
Σ(P )
− k-face f
P ,aasdasdasdqweqw
Σ(P )
− k-face f
aasdasdasdqweqw − all facets f1, . . . , fd−k containing f
P ,aasdasdasdqweqw
Σ(P )
− k-face f
aasdasdasdqweqw − all facets f1, . . . , fd−k containing f
aasdasdasdqweqw − corresponding reflections generate a finite group
P ,aasdasdasdqweqw
Σ(P )
− k-face f
aasdasdasdqweqw − all facets f1, . . . , fd−k containing f
aasdasdasdqweqw − corresponding reflections generate a finite group
aasdasdasdqweqw − corresponding nodes span elliptic subdiagram
P ,aasdasdasdqweqw
Σ(P )
− k-face f
aasdasdasdqweqw − all facets f1, . . . , fd−k containing f
aasdasdasdqweqw − corresponding reflections generate a finite group
aasdasdasdqweqw − corresponding nodes span elliptic subdiagram
− elliptic subdiagram spanned by nodes n1, . . . , nd−k
P ,aasdasdasdqweqw
Σ(P )
− k-face f
aasdasdasdqweqw − all facets f1, . . . , fd−k containing f
aasdasdasdqweqw − corresponding reflections generate a finite group
aasdasdasdqweqw − corresponding nodes span elliptic subdiagram
− elliptic subdiagram spanned by nodes n1, . . . , nd−k
− vectors v1, . . . , vd−k orthogonal to corresponding facets f1, . . . , fd−k ;
asdaa v1, . . . , vd−k span positive definite subspace
P ,aasdasdasdqweqw
Σ(P )
− k-face f
aasdasdasdqweqw − all facets f1, . . . , fd−k containing f
aasdasdasdqweqw − corresponding reflections generate a finite group
aasdasdasdqweqw − corresponding nodes span elliptic subdiagram
− elliptic subdiagram spanned by nodes n1, . . . , nd−k
− vectors v1, . . . , vd−k orthogonal to corresponding facets f1, . . . , fd−k ;
asdaa v1, . . . , vd−k span positive definite subspace
− hv1, . . . vd−k i⊥ is a k-dimensional hyperbolic plane Πk
P ,aasdasdasdqweqw
Σ(P )
− k-face f
aasdasdasdqweqw − all facets f1, . . . , fd−k containing f
aasdasdasdqweqw − corresponding reflections generate a finite group
aasdasdasdqweqw − corresponding nodes span elliptic subdiagram
− elliptic subdiagram spanned by nodes n1, . . . , nd−k
− vectors v1, . . . , vd−k orthogonal to corresponding facets f1, . . . , fd−k ;
asdaa v1, . . . , vd−k span positive definite subspace
− hv1, . . . vd−k i⊥ is a k-dimensional hyperbolic plane Πk
− Πk is preserved by all reflections with resp. to to fi.
P ,aasdasdasdqweqw
Σ(P )
− k-face f
aasdasdasdqweqw − all facets f1, . . . , fd−k containing f
aasdasdasdqweqw − corresponding reflections generate a finite group
aasdasdasdqweqw − corresponding nodes span elliptic subdiagram
− elliptic subdiagram spanned by nodes n1, . . . , nd−k
− vectors v1, . . . , vd−k orthogonal to corresponding facets f1, . . . , fd−k ;
asdaa v1, . . . , vd−k span positive definite subspace
− hv1, . . . vd−k i⊥ is a k-dimensional hyperbolic plane Πk
− Πk is preserved by all reflections with resp. to f1, . . . , fd−k
− ∩fi = Πk contains a k-face f
• If P is compact then P is simple.
• If P is compact (finite volume) then P is indecomposable.
• Coxeter diagram → combinatorics of P .
• If P is compact then P is simple.
• If P is compact (finite volume) then P is indecomposable.
• Coxeter diagram → combinatorics of P .
− k-faces ↔ elliptic subdiagrams of order d − k,
( elliptic = Coxeter diagrams of spherical simplices).
− vertices at ∂Hd ↔ parabolic subdiagrams of order d
( parabolic = Coxeter diagrams of Euclidean simplices).
− Finite volume ↔ P comb. equiv. to a Euclidean polytope
• If P is compact then P is simple.
• If P is compact (finite volume) then P is indecomposable.
• Coxeter diagram → combinatorics of P .
− k-faces ↔ elliptic subdiagrams of order d − k,
( elliptic = Coxeter diagrams of spherical simplices).
− vertices at ∂Hd ↔ parabolic subdiagrams of order d
( parabolic = Coxeter diagrams of Euclidean simplices).
− Finite volume ↔ P comb. equiv. to a Euclidean polytope
Thm. (Vinberg ’67)
Indecomposable, symmetric matrix G, sgn(G) = (d, 1),
gii = 1,
gij ≤ 0.
Then there exists a convex polytope P ⊂ Hd, such that G = G(P )
(unique up to an isometry of Hd).
Thm. (Vinberg ’67)
Indecomposable, symmetric matrix G, sgn(G) = (d, 1),
gii = 1,
gij ≤ 0.
Then there exists a convex polytope P ⊂ Hd, such that G = G(P )
(unique up to an isometry of Hd).
• P is compact (finite volume) ⇒ P combinatorially equivalent to
some Euclidean polytope.
Thm. (Vinberg ’67)
Indecomposable, symmetric matrix G, sgn(G) = (d, 1),
gii = 1,
gij ≤ 0.
Then there exists a convex polytope P ⊂ Hd, such that G = G(P )
(unique up to an isometry of Hd).
• P is compact (finite volume) ⇒ P combinatorially equivalent to
some Euclidean polytope.
1. P has at least one vertex (ideal vertex).
2. each edge has two ends.
Thm. (Vinberg ’67)
Indecomposable, symmetric matrix G, sgn(G) = (d, 1),
gii = 1,
gij ≤ 0.
Then there exists a convex polytope P ⊂ Hd, such that G = G(P )
(unique up to an isometry of Hd).
• P is compact (finite volume) ⇒ P combinatorially equivalent to
some Euclidean polytope.
1. P has at least one vertex (ideal vertex).
2. each edge has two ends.
• 1, 2 ⇒ P is compact (finite volume).
Thm. (Vinberg ’84) If P ⊂ Hd is compact then d ≤ 29.
Thm. (Vinberg ’84) If P ⊂ Hd is compact then d ≤ 29.
Idea of proof:
1. vertices ↔ elliptic subdiagrams ↔ many right angles
2. triangular, quadrilateral faces ↔ many non-right angles
Thm. (Vinberg ’84) If P ⊂ Hd is compact then d ≤ 29.
Idea of proof:
1. vertices ↔ elliptic subdiagrams ↔ many right angles
2. triangular, quadrilateral faces ↔ many non-right angles
3. Thm. (Nikulin, 81):
For any simple, compact, convex polytope P ⊂ Ed
and any i < k ≤ [d/2] holds
αki
<
d−i
d−k
[d/2]
i
[d/2]
k
+
+
[(d+1)/2]
i
[(d+1)/2]
k
where αki = average number of i-faces of a k-face of P
Thm. (Vinberg ’84) If P ⊂ Hd is compact then d ≤ 29.
Idea of proof:
1. vertices ↔ elliptic subdiagrams ↔ many right angles
2. triangular, quadrilateral faces ↔ many non-right angles
3. a2 ≤
4(d−ε)
(d−1−ε)
a2=
average number of sides of 2-face
ε=
1, if d is even
0, otherwise.
Thm. (Vinberg ’84) If P ⊂ Hd is compact then d ≤ 29.
Idea of proof:
1. vertices ↔ elliptic subdiagrams ↔ many right angles
2. triangular, quadrilateral faces ↔ many non-right angles
3. a2 ≤
4(d−ε)
(d−1−ε)
werwer ⇒
a2=
average number of sides of 2-face
ε=
1, if d is even
0, otherwise.
a lots of triangular and quadrilateral 2-faces
More precisely:
Plane angles −→ weights
P
vertex A −→ σ(A) =
of weights of plane angles at A
P
2-face F −→ σ(F ) =
of weights of plane angles of F
L. If for all A, F σ(A) ≤ cd and σ(F ) ≥ 5 − nF then d < 8c + 6.
(nF = # of sides of F )
plane angle ↔ diagram ΣA of a vertex A with two “black” nodes a
and b (corresp. to facets not containing F ).
weight = 1, if distΣA (a, b) ≤ 7
weight = 0, otherwise.
• If P ⊂ Hd is compact then d ≤ 29. (Vinberg ’84).
Examples known for d ≤ 8.
Unique Ex. for d = 8 (Bugaenko ’92):
• If P ⊂ Hd is compact then d ≤ 29. (Vinberg ’84).
Examples known for d ≤ 8.
Unique Ex. for d = 8 (Bugaenko ’92):
All known Ex. for d = 7 (Bugaenko ’84):
• If P ⊂ Hd is compact then d ≤ 29. (Vinberg ’84).
Examples known for d ≤ 8.
Unique Ex. for d = 8 (Bugaenko ’92):
All known Ex. for d = 7 (Bugaenko ’84):
• If P ⊂ Hd is compact then d ≤ 29. (Vinberg ’84).
Examples known for d ≤ 8.
Unique Ex. for d = 8 (Bugaenko ’92):
• If P ⊂ Hd is of finite volume then d < 996.
(Prokhorov, Khovanskii ’86).
• If P ⊂ Hd is compact then d ≤ 29. (Vinberg ’84).
Examples known for d ≤ 8.
Unique Ex. for d = 8 (Bugaenko ’92):
• If P ⊂ Hd is of finite volume then d < 996.
(Prokhorov, Khovanskii ’86).
Examples known for d ≤ 19 (Vinberg, Kaplinskaya ’78)
d = 21 (Borcherds ’87).
Example: Compact right-angled polytopes
(Vinberg, Potyagailo ’05)
Example: Compact right-angled polytopes
(Vinberg, Potyagailo ’05)
• Each face is a right-angled polytope.
Example: Compact right-angled polytopes
(Vinberg, Potyagailo ’05)
• Each face is a right-angled polytope.
• Each 2-face has at least 5 sides.
Example: Compact right-angled polytopes
(Vinberg, Potyagailo ’05)
• Each face is a right-angled polytope.
• Each 2-face has at least 5 sides.
• a2 ≤
4(d−ε)
(d−1−ε)
⇒ d ≤ 4 (for compact polytopes).
Example: Compact right-angled polytopes
(Vinberg, Potyagailo ’05)
• Each face is a right-angled polytope.
• Each 2-face has at least 5 sides.
• a2 ≤
4(d−ε)
(d−1−ε)
⇒ d ≤ 4 (for compact polytopes).
•
d=2
d=3
d=4
regular 4-polytope
with 120 dodecahedral
facets
Example: Compact right-angled polytopes
(Vinberg, Potyagailo ’05)
• Each face is a right-angled polytope.
• Each 2-face has at least 5 sides.
• a2 ≤
4(d−ε)
(d−1−ε)
⇒ d ≤ 4 (for compact polytopes).
•
d=2
d=3
d=4
regular 4-polytope
with 120 dodecahedral
facets
Example: Finite volume right-angled polytopes
(Vinberg, Potyagailo ’05)
• d ≤ 14
• Examples are known up to d = 8 only
(from rotations around vertices of simplices):
Example: Finite volume right-angled polytopes
(Vinberg, Potyagailo ’05)
• d ≤ 14
• Examples are known up to d = 8 only
(from rotations around vertices of simplices):
Example: 2-Polytopes (polygons).
Example: 2-Polytopes (polygons).
P
• Poincare (1882):
αi < π(n − 2).
Example: 2-Polytopes (polygons).
P
• Poincare (1882):
αi < π(n − 2).
• n-gon with fixed angles
depends on n − 3 continuous parameters.
Example: 2-Polytopes (polygons).
P
• Poincare (1882):
αi < π(n − 2).
• n-gon with fixed angles
depends on n − 3 continuous parameters.
Thm. (Andreev ’70):
Compact acute-angled polytope in Hd, d ≥ 3
is determined (up to isometry) by
its combinatorial type and dihedral angles.
Example: 3-Polytopes.
Example: 3-Polytopes.
Thm. (Andreev ’70). Given a combinatorial type of a simple
3-polytope and prescribed acute dihedral angles, the polytope is
realized by a compact polytope in H3 if and only if
Example: 3-Polytopes.
Thm. (Andreev ’70). Given a combinatorial type of a simple
3-polytope and prescribed acute dihedral angles, the polytope is
realized by a compact polytope in H3 if and only if
•
β
α
γ
β
α+β+γ >π
α
γ
β
α+β+γ <π
α
δ
γ
α + β + γ + δ < 2π
Example: 3-Polytopes.
Thm. (Andreev ’70). Given a combinatorial type of a simple
3-polytope and prescribed acute dihedral angles, the polytope is
realized by a compact polytope in H3 if and only if
•
β
α
γ
β
α+β+γ >π
α
γ
β
α+β+γ <π
α
δ
γ
α + β + γ + δ < 2π
ϕ6
• For a simplex: det(G(P )) < 0.
• For a triangular prism: ∃i ∈ {1, 2, . . . , 6}: ϕi 6= π/2
ϕ5
ϕ3
ϕ4
ϕ2
ϕ1
Example: 3-Polytopes.
Thm. (Andreev ’70). Given a combinatorial type of a simple
3-polytope and prescribed acute dihedral angles, the polytope is
realized by a compact polytope in H3 if and only if
•
β
α
γ
β
α+β+γ >π
α
γ
β
α+β+γ <π
α
δ
γ
α + β + γ + δ < 2π
ϕ6
• For a simplex: det(G(P )) < 0.
• For a triangular prism: ∃i ∈ {1, 2, . . . , 6}: ϕi 6= π/2
ϕ5
ϕ3
ϕ4
ϕ2
ϕ1
Example: 3-Polytopes.
Thm. (Andreev ’70). Given a combinatorial type of a simple
3-polytope and prescribed acute dihedral angles, the polytope is
realized by a compact polytope in H3 if and only if
•
β
α
γ
β
α+β+γ >π
α
γ
β
α+β+γ <π
α
δ
γ
α + β + γ + δ < 2π
ϕ6
• For a simplex: det(G(P )) < 0.
• For a triangular prism: ∃i ∈ {1, 2, . . . , 6}: ϕi 6= π/2
ϕ5
ϕ3
ϕ4
ϕ2
ϕ1
Example: 3-Polytopes.
Thm. (Andreev ’70). Given a combinatorial type of a simple
3-polytope and prescribed acute dihedral angles, the polytope is
realized by a compact polytope in H3 if and only if
•
β
α
γ
β
α+β+γ >π
α
γ
β
α+β+γ <π
α
δ
γ
α + β + γ + δ < 2π
ϕ6
• For a simplex: det(G(P )) < 0.
• For a triangular prism: ∃i ∈ {1, 2, . . . , 6}: ϕi 6= π/2
ϕ5
ϕ3
ϕ4
ϕ2
ϕ1
Additional conditions for finite volume polytopes:
•
ϕ1
ϕ4
ϕ3
ϕ2
⇒
ϕi = π2 , i = 1, . . . , 4.
⇒
either α 6=
•
β
α
π
2
or β 6= π2 .
Example: no angles of this polytope
would satisfy the conditions of the theorem!
Thm. (Andreev ’70) Let P be an acute-angled polytope in Hd,
a, b be its faces, and ā, b̄ be planes spanned by a and b.
If a ∩ b = ∅ then ā ∩ b̄ = ∅.
Hyperbolic
reflection
groups
Compact hyperbolic Coxeter polytopes
1. By dimension.
Compact hyperbolic Coxeter polytopes
1. By dimension.
• dim = 2. Poincare (1882):
P
αi < π(n − 2).
Compact hyperbolic Coxeter polytopes
1. By dimension.
• dim = 2. Poincare (1882):
P
αi < π(n − 2).
• dim = 3. Andreev (’70): necessary and suff. condition
for dihedral angles.
Compact hyperbolic Coxeter polytopes
1. By dimension.
• dim = 2. Poincare (1882):
P
αi < π(n − 2).
• dim = 3. Andreev (’70): necessary and suff. condition
for dihedral angles.
• dim ≥ 4. ?????
2. By number of facets.
2. By number of facets.
• n = d + 1, simplices (Lannér ’52)
2. By number of facets.
• n = d + 1, simplices (Lannér ’52), Lannér diagrams
k
l
m
1
k
+
1
l
+
1
m
<1
2. By number of facets.
• n = d + 1, simplices (Lannér ’52): d ≤ 4, fin. many for d > 2.
2. By number of facets.
• n = d + 1, simplices (Lannér ’52): d ≤ 4, fin. many for d > 2.
• n = d + 2, ∆k × ∆l
− prisms (Kaplinskaja ’74): d ≤ 5, fin. many for d > 3.
− others (Esselmann ’96): d = 4, ∆2 × ∆2, 7 items.
2. By number of facets.
• n = d + 1, simplices (Lannér ’52): d ≤ 4, fin. many for d > 2.
• n = d + 2, ∆k × ∆l
− prisms (Kaplinskaja ’74): d ≤ 5, fin. many for d > 3.
− others (Esselmann ’96): d = 4, ∆2 × ∆2, 7 items.
Examples of prisms:
2. By number of facets.
• n = d + 1, simplices (Lannér ’52): d ≤ 4, fin. many for d > 2.
• n = d + 2, ∆k × ∆l
− prisms (Kaplinskaja ’74): d ≤ 5, fin. many for d > 3.
− others (Esselmann ’96): d = 4, ∆2 × ∆2, 7 items.
Examples of prisms:
Esselmann’s polytopes:
10
10
8
8
8
2. By number of facets.
• n = d + 1, simplices (Lannér ’52): d ≤ 4, fin. many for d > 2.
• n = d + 2, ∆k × ∆l
− prisms (Kaplinskaja ’74): d ≤ 5, fin. many for d > 3.
− others (Esselmann ’96): d = 4, ∆2 × ∆2, 7 items.
• n = d + 3, several combinatorial types
(Tumarkin ’03): d ≤ 6 or d = 8, fin. many for d > 3.
2. By number of facets.
• n = d + 1, simplices (Lannér ’52): d ≤ 4, fin. many for d > 2.
• n = d + 2, ∆k × ∆l
− prisms (Kaplinskaja ’74): d ≤ 5, fin. many for d > 3.
− others (Esselmann ’96): d = 4, ∆2 × ∆2, 7 items.
• n = d + 3, several combinatorial types
(Tumarkin ’03): d ≤ 6 or d = 8, fin. many for d > 3.
8
Examples:
8
Examples: compact polytopes with any number of facets
in H3, H4, H5 and H6.
Examples: compact polytopes with any number of facets
in H3, H4, H5 and H6.
8
8
Examples: compact polytopes with any number of facets
in H3, H4, H5 and H6.
8
8
Examples: compact polytopes with any number of facets
in H3, H4, H5 and H6.
8
8
Examples: compact polytopes with any number of facets
in H3, H4, H5 and H6.
8
8
Examples: compact polytopes with any number of facets
in H3, H4, H5 and H6.
8
8
2. By number of facets.
• n = d + 1, simplices (Lannér ’52): d ≤ 4, fin. many for d > 2.
• n = d + 2, ∆k × ∆l
− prisms (Kaplinskaja ’74): d ≤ 5, fin. many for d > 3.
− others (Esselmann ’96): d = 4, ∆2 × ∆2, 7 items.
• n = d + 3, several combinatorial types
(Tumarkin ’03): d ≤ 6 or d = 8, fin. many for d > 3.
• n = d + 4, really many combinatorial types...
?????
2. By number of facets.
• n = d + 1, simplices (Lannér ’52): d ≤ 4, fin. many for d > 2.
• n = d + 2, ∆k × ∆l
− prisms (Kaplinskaja ’74): d ≤ 5, fin. many for d > 3.
− others (Esselmann ’96): d = 4, ∆2 × ∆2, 7 items.
• n = d + 3, several combinatorial types
(Tumarkin ’03): d ≤ 6 or d = 8, fin. many for d > 3.
• n = d + 4, really many combinatorial types...
How to proceed for a given combinatorial type ?
How to list all appropriate combinatorial types ?
Tools
• Given a combinatorial type, may try to “reconstruct” the polytope
(i.e. to find its dihedral angles).
Combinatorics:
Diagram of missing faces
Dihedral angles:
Coxeter diagram
Diagram of missing faces
• Nodes ←→ facets of P
• Missing face is a minimal set of facets f1, ..., fk ,
Tk
such that i=1 fi = ∅.
• Missing faces are encircled.
• Ex:
Tools
• Given a combinatorial type, may try to “reconstruct” the polytope
(i.e. to find its dihedral angles).
Combinatorics:
Diagram of missing faces
Missing faces
Dihedral angles:
Coxeter diagram
←→
Lannér subdiagrams
(minimal non-elliptic subd.)
Tools
• Given a combinatorial type, may try to “reconstruct” the polytope
(i.e. to find its dihedral angles).
Combinatorics:
Diagram of missing faces
Missing faces
Example:
Dihedral angles:
Coxeter diagram
←→
Lannér subdiagrams
(minimal non-elliptic subd.)
Tools
• Given a combinatorial type, may try to “reconstruct” the polytope
(i.e. to find its dihedral angles).
Combinatorics:
Diagram of missing faces
Missing faces
Example:
Dihedral angles:
Coxeter diagram
←→
Lannér subdiagrams
(minimal non-elliptic subd.)
Tools
• Given a combinatorial type, may try to “reconstruct” the polytope
(i.e. to find its dihedral angles).
Combinatorics:
Diagram of missing faces
Missing faces
Example:
Dihedral angles:
Coxeter diagram
←→
Lannér subdiagrams
(minimal non-elliptic subd.)
Lannér subdiagrams ←→ Missing faces
• If L is a Lannér diagram then |L| ≤ 5.
• # of Lannér diagrams of order 4, 5 is finite.
• For any two Lannér subdiagrams s.t. L1 ∩ L2 = ∅,
there exists an edge joining these subdiagrams.
Given a combinatorial type may try to check
if there is a Coxeter polytope of this type.
Tools
• Combinatorial type → “reconstruction” of Coxeter polytope
• Coxeter faces.
− Borcherds ’87:
Elliptic subdiagram
without An and D5
→ Coxeter face
Tools
• Combinatorial type → “reconstruction” of Coxeter polytope
• Coxeter faces.
− Borcherds ’87:
Elliptic subdiagram
without An and D5
− Allcock ’05:
Angles of this face are easy to find.
→ Coxeter face
Tools
• Combinatorial type → “reconstruction” of Coxeter polytope
• Coxeter faces.
− Borcherds ’87:
Elliptic subdiagram
without An and D5
− Allcock ’05:
Angles of this face are easy to find.
Examples:
→ Coxeter face
Tools
• Combinatorial type → “reconstruction” of Coxeter polytope
• Coxeter faces.
− Borcherds ’87:
Elliptic subdiagram
without An and D5
− Allcock ’05:
Angles of this face are easy to find.
Examples:
→ Coxeter face
Tools
• Combinatorial type → “reconstruction” of Coxeter polytope
• Coxeter faces.
− Borcherds ’87:
Elliptic subdiagram
without An and D5
− Allcock ’05:
Angles of this face are easy to find.
Examples:
→ Coxeter face
Tools
• Combinatorial type → “reconstruction” of Coxeter polytope
• Coxeter faces.
− Borcherds ’87:
Elliptic subdiagram
without An and D5
− Allcock ’05:
Angles of this face are easy to find.
Examples:
→ Coxeter face
Tools
• Combinatorial type → “reconstruction” of Coxeter polytope
• Coxeter faces.
− Borcherds ’87:
Elliptic subdiagram
without An and D5
− Allcock ’05:
Angles of this face are easy to find.
Examples:
→ Coxeter face
Tools
• Combinatorial type → “reconstruction” of Coxeter polytope
• Coxeter faces.
− Borcherds ’87:
Elliptic subdiagram
without An and D5
− Allcock ’05:
Angles of this face are easy to find.
Examples:
→ Coxeter face
Tools
• Combinatorial type → “reconstruction” of Coxeter polytope
• Coxeter faces.
− Borcherds ’87:
Elliptic subdiagram
without An and D5
− Allcock ’05:
Angles of this face are easy to find.
Examples:
10
→ Coxeter face
Tools
• Combinatorial type → “reconstruction” of Coxeter polytope
• Coxeter faces.
− Borcherds ’87:
Elliptic subdiagram
without An and D5
− Allcock ’05:
Angles of this face are easy to find.
Examples:
10
→ Coxeter face
Tools
• Combinatorial type → “reconstruction” of Coxeter polytope
• Coxeter faces.
− Borcherds ’87:
Elliptic subdiagram
without An and D5
− Allcock ’05:
Angles of this face are easy to find.
Examples:
10
→ Coxeter face
Tools
• Combinatorial type → “reconstruction” of Coxeter polytope
• Coxeter faces.
− Borcherds ’87:
Elliptic subdiagram
without An and D5
− Allcock ’05:
Angles of this face are easy to find.
Examples:
10
→ Coxeter face
Tools
• Combinatorial type → “reconstruction” of Coxeter polytope
• Coxeter faces.
− Borcherds ’87:
Elliptic subdiagram
without An and D5
− Allcock ’05:
Angles of this face are easy to find.
Examples:
10
→ Coxeter face
Tools
• Combinatorial type → “reconstruction” of Coxeter polytope
• Coxeter faces.
• det(G(P )) = 0.
Tools
• Combinatorial type → “reconstruction” of Coxeter polytope
• Coxeter faces.
• det(G(P )) = 0.
• Local determinants:
det(Σ, T ) =
detΣ
det(Σ\T ) .
Tools
• Combinatorial type → “reconstruction” of Coxeter polytope
• Coxeter faces.
• det(G(P )) = 0.
• Local determinants:
det(Σ, T ) =
detΣ
det(Σ\T ) .
det(Σ, v) = det(S1, v) + det(S2, v) − 1
det(Σ, hv1, v2i) = det(S1, v1)det(S2, v2) −
S1
2
g12
S1
v
v1 v 2
S2
S2
Tools
• Combinatorial type → “reconstruction” of Coxeter polytope
• Coxeter faces.
• det(G(P )) = 0.
• Local determinants.
To list combinatorial types:
• Gale diagram (works well for n ≤ d + 3 only).
Tools
• Combinatorial type → “reconstruction” of Coxeter polytope
• Coxeter faces.
• det(G(P )) = 0.
• Local determinants.
To list combinatorial types:
• Gale diagram (works well for n ≤ d + 3 only).
◦ ∀u ∈ Σ(P ) ∃ Lannér subdiagram L, u ∈ L.
◦ ∀ Lannér subdiagram L1 ∃ Lan. subd. L2, L1 ∩ L2 = ∅.
2. By number of facets.
• n = d + 1, simplices (Lannér ’52): d ≤ 4, fin. many for d > 2.
• n = d + 2, ∆k × ∆l
− prisms (Kaplinskaja ’74): d ≤ 5, fin. many for d > 3.
− others (Esselmann ’96): d = 4, ∆2 × ∆2, 7 items.
• n = d + 3, many combinatorial types
(Tumarkin ’03): d ≤ 6 or d = 8, fin.many for d > 3.
• n = d + 4, really many combinatorial types...
(T,F ’06): d ≤ 7, unique example in d = 7.
• n = d + 5, (T,F ’06): d ≤ 8.
1
d+1
d+2
d+3
d+4
d+5
n
2
3
4
5
6
7
8
dim
1
d+1
d+2
d+3
d+4
d+5
n
2
3
4
5
6
7
8
29
dim
1
d+1
d+2
d+3
d+4
d+5
n
2
3
4
5
6
7
8
29
dim
1
d+1
d+2
d+3
d+4
d+5
n
2
3
4
5
6
7
8
29
dim
1
d+1
d+2
d+3
d+4
d+5
n
2
3
4
5
6
7
8
29
dim
1
d+1
d+2
d+3
d+4
d+5
n
2
3
4
5
6
7
8
29
dim
1
d+1
d+2
d+3
d+4
d+5
n
2
3
4
5
6
7
8
29
dim
1
d+1
d+2
d+3
d+4
d+5
n
2
3
4
5
6
7
8
29
dim
1
d+1
d+2
d+3
d+4
d+5
n
2
3
4
5
6
7
8
29
dim
1
d+1
d+2
d+3
d+4
d+5
n
2
3
4
5
6
7
8
29
dim
1
2
3
4
5
6
7
8
29
d+1
d+2
d+3
d+4
d+5
n
1) proofs are similar
2) use previous cases
dim
1
2
3
4
5
6
7
8
29
d+1
d+2
d+3
d+4
d+5
n
1) proofs are similar
2) use previous cases
Inductive algorithm?
dim
3. By number of dotted edges.
3. By number of dotted edges.
• p = 0, (T,F ’06): Simplices and Esselmann’s polytopes only.
d ≤ 4, n ≤ d + 2.
3. By number of dotted edges.
• p = 0, (T,F ’06): Simplices and Esselmann’s polytopes only.
d ≤ 4, n ≤ d + 2.
• p = 1, (T,F ’07): Only polytopes with n ≤ d + 3.
d ≤ 6 and d = 8.
3. By number of dotted edges.
• p = 0, (T,F ’06): Simplices and Esselmann’s polytopes only.
d ≤ 4, n ≤ d + 2.
• p = 1, (T,F ’07): Only polytopes with n ≤ d + 3.
d ≤ 6 and d = 8.
• p ≤ n − d − 2, (T,F ’07): finitely many polytopes. Algorithm.
3. By number of dotted edges.
• p = 0, (T,F ’06): Simplices and Esselmann’s polytopes only.
d ≤ 4, n ≤ d + 2.
• p = 1, (T,F ’07): Only polytopes with n ≤ d + 3.
d ≤ 6 and d = 8.
• p ≤ n − d − 2, (T,F ’07): finitely many polytopes. Algorithm.
• (T,F ’06): If all Lannér subdiagrams are of order 2, then d ≤ 13.
(for compact or simple finite volume polytopes).
Finite volume polytopes
• combinatorics: not “simple” but “simple in edges”
(a k-face is contained in d − k facets unless k = 0).
• missing face ↔ Lannér or quasi-Lannér subdiagram
(i.e. diagram of a simplex with some vertices at ∂Hd).
• n = d + 1, simplices. d ≤ 9, fin. many for d ≥ 3.
• n = d + 2, ◦ ∆i × ∆j ⇒ · prisms, d ≤ 5, fin. many for d > 3.
· ∆2 × ∆2 ⇒
◦ pyramid over ∆i×∆j , d = 3, . . . , 13 and 17(fin. many).
• n = d + 3, d ≤ 16. A unique example in d = 16:
Reflection subgroups of reflection groups
Reflection subgroups of reflection groups
• A standard subgroup of a reflection group G = {ri | ...} is a
reflection group generated by some of ri.
Reflection subgroups of reflection groups
• A standard subgroup of a reflection group G = {ri | ...} is a
reflection group generated by some of ri.
Thm. (Deodhar ’82). Let G be an infinite indecomposable Coxeter
group and H ⊂ G be a proper standard subgroup of G. Then
[G : H] = ∞.
Reflection subgroups of reflection groups
• A standard subgroup of a reflection group G = {ri | ...} is a
reflection group generated by some of ri.
Thm. (Deodhar ’82). Let G be an infinite indecomposable Coxeter
group and H ⊂ G be a proper standard subgroup of G. Then
[G : H] = ∞.
Question. What about finite index reflection subgroups?
Reflection subgroups of reflection groups
• A standard subgroup of a reflection group G = {ri | ...} is a
reflection group generated by some of ri.
Thm. (Deodhar ’82). Let G be an infinite indecomposable Coxeter
group and H ⊂ G be a proper standard subgroup of G. Then
[G : H] = ∞.
Question. What about finite index reflection subgroups?
aaaaaaaa ”yes” ↔
tiling of a Coxeter polytope
by Coxeter polytopes.
Reflection subgroups of reflection groups
Example:
:
( π7 , π7 , π7 ) tiled by 24 copies of ( π2 , π3 , π7 )
Reflection subgroups of reflection groups
Example:
( π7 , π7 , π7 ) tiled by 24 copies of ( π2 , π3 , π7 )
:
Example: A reflection group generated by ( π5 , π5 , π5 ) has no finite
index reflection subgroups.
Reflection subgroups of reflection groups
Example:
( π7 , π7 , π7 ) tiled by 24 copies of ( π2 , π3 , π7 )
:
Example: A reflection group generated by ( π5 , π5 , π5 ) has no finite
index reflection subgroups.
Thm. (T,F ’03) G infinite indecomposable group, H ⊂ G a finite
index reflection subgroup. Then rk H ≥ rk G.
(rk G is a number of reflections generating G).