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Local rigidityof hyperbolicright-angledCoxetergroups in
dimension 4and 5
TomoshigeYukita
Contents
Section 1
Section 2
Local rigidity of hyperbolic right-angledCoxeter groups in
dimension 4 and 5
Tomoshige Yukita
Waseda University
Topology and Computer 2019, Osaka City UniversityOctober 19,
2019
1 / 23
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Local rigidityof hyperbolicright-angledCoxetergroups in
dimension 4and 5
TomoshigeYukita
Contents
Section 1
Section 2
Contents
1. Definitions and main result
2. Idea of proof
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Local rigidityof hyperbolicright-angledCoxetergroups in
dimension 4and 5
TomoshigeYukita
Contents
Section 1
Section 2
Section 1Definitions and main result
3 / 23
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Local rigidityof hyperbolicright-angledCoxetergroups in
dimension 4and 5
TomoshigeYukita
Contents
Section 1
Section 2
Main result
Theorem (Y)
There exists a right-angled 5-polytope P of finite-volumesuch
that all the RACGs with Fuchsian ends obtained fromΓP are locally
rigid.
Idea
(How to construct)Using vertical projection from ∞ and the
computer programCoxIter (R.Guglielmetti, 2015).
(How to verify the local rigidity)Using rigidity of the shapes
of the link of ideal vertices of P.
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Local rigidityof hyperbolicright-angledCoxetergroups in
dimension 4and 5
TomoshigeYukita
Contents
Section 1
Section 2
Hyperbolic d-polytope
・Ud : the upper-half space model of hyperbolic
d-space.Definition (Hyperbolic d-polytope)
・P = ∩Ni=1H−i : a hyperbolic d -polytope.・P: a hyperbolic
Coxeter d-polytopedef⇔ its dihedral angles = πm (m ≥ 2 or ∞).・P: a
hyperbolic right-angled d-polytopedef⇔ its dihedral angles = π2
.
・P: finite volume ⇒ P ∩ ∂Ud = {v1, · · · , vk}: ideal
vertices.
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Local rigidityof hyperbolicright-angledCoxetergroups in
dimension 4and 5
TomoshigeYukita
Contents
Section 1
Section 2
d-dimensional hyperbolic Coxeter group
・ri ∈ Isom(Ud): the reflection in the bounding hyperplane Hi・S
:= { r1, · · · , rN }.Definition (d-dimensional hyperbolic Coxeter
group)
ΓP := ⟨S⟩: d -dim. hyp. Coxeter group associated with P.If P is
right-angled,
ΓP : d -dim. hyp. RACG associated with P.
・ΓP < Isom(Ud): a discrete subgroup.・ΓP has the following
nice presentation:
ΓP =⟨r1, · · · , rN | r2k = 1, (ri rj)mij = 1
⟩where
π
mij= the dihedral angle between Fi and Fj
mij = ∞ if Fi and Fj do not intersect6 / 23
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Local rigidityof hyperbolicright-angledCoxetergroups in
dimension 4and 5
TomoshigeYukita
Contents
Section 1
Section 2
An example of hyperbolic Coxeter group
7 / 23
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Local rigidityof hyperbolicright-angledCoxetergroups in
dimension 4and 5
TomoshigeYukita
Contents
Section 1
Section 2
Main result
Theorem (Y)
There exists a right-angled 5-polytope P of finite-volumesuch
that all the RACGs with Fuchsian ends obtained fromΓP are locally
rigid.
Idea
(How to construct)Using vertical projection from ∞ and the
computer programCoxIter (R.Guglielmetti, 2015).
(How to verify the local rigidity)Using rigidity of the shapes
of the link of ideal vertices of P.
8 / 23
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Local rigidityof hyperbolicright-angledCoxetergroups in
dimension 4and 5
TomoshigeYukita
Contents
Section 1
Section 2
RACGs with Fuchsian ends
・P: a hyperbolic right-angled d-polytope of finite volume・Γ: the
RACG associated with P (finite-covolume).・P ′: the d-polytope
obtained by removing mutually disjointfacets F1, · · · ,Fk of
PThen, the RACG Γ′ associated with P ′ is said to be withFuchsian
ends.
9 / 23
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Local rigidityof hyperbolicright-angledCoxetergroups in
dimension 4and 5
TomoshigeYukita
Contents
Section 1
Section 2
Main result
Theorem (Y)
There exists a right-angled 5-polytope P of finite-volumesuch
that all the RACGs with Fuchsian ends obtained fromΓP are locally
rigid.
Idea
(How to construct)Using vertical projection from ∞ and the
computer programCoxIter (R.Guglielmetti, 2015).
(How to verify the local rigidity)Using rigidity of the shapes
of the link of ideal vertices of P.
10 / 23
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Local rigidityof hyperbolicright-angledCoxetergroups in
dimension 4and 5
TomoshigeYukita
Contents
Section 1
Section 2
Representation space and local rigidity
・Gd := Isom(Ud): the isometry group of Ud .・Γ < Gd : a
discrete subgroup. (↔ Ud/Γ: a hyp. orbifold)・ρ0 : Γ ↪→ Gd : the
inclusion map.Definition
R(Γ,Gd ): the space of all homomorphisms from Γ to Gd
withtopology of pointwise-convergence.
・Gd ↷ R(Γ,Gd) by conjugation; (gρ)(γ) = gρ(γ)g−1.
Definition
ρ : I → R(Γ,Gd): trivialdef⇔ ∃gt ∈ Gd (t ∈ I ) s.t. ρt =
gtρ0.
・If ΓP is Coxeter group, thenρ : I → R(ΓP ,Gd) ↔ moving the
facets of P.
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Local rigidityof hyperbolicright-angledCoxetergroups in
dimension 4and 5
TomoshigeYukita
Contents
Section 1
Section 2
An example (not locally rigid)
・O ⊂ U3: the regular ideal right-angled octahedron.⇝ ΓO has
non-trivial deformation path (by Andreev’s theorem).
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Local rigidityof hyperbolicright-angledCoxetergroups in
dimension 4and 5
TomoshigeYukita
Contents
Section 1
Section 2
Local rigidity
Definition
Γ is locally rigiddef⇔ any representation ρ ∈ R(Γ,Gd) near
by
ρ0 is obtained by conjugation, that is, there is an open n.b.d
ofρ0 contained in the orbit Gd · ρ0
Γ : locally rigid ⇒ any ρ : I → R(Γ,Gd) : trivial (near by ρ0)⇒
the orbifold Ud/Γ can not be deformed.
Question
When is a discrete subgroup Γ locally rigid or not?
・For cocompact subgroups (Calabi, 1961).・For cofinite subgroups
(Garland-Raghunathan, 1970).
13 / 23
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Local rigidityof hyperbolicright-angledCoxetergroups in
dimension 4and 5
TomoshigeYukita
Contents
Section 1
Section 2
Main result
Theorem (Y)
There exists a right-angled 5-polytope P of finite-volumesuch
that all the RACGs with Fuchsian ends obtained fromΓP are locally
rigid.
Idea
(How to construct)Using vertical projection from ∞ and the
computer programCoxIter (R.Guglielmetti, 2015).
(How to verify the local rigidity)Using rigidity of the shapes
of the link of ideal vertices of P.
14 / 23
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Local rigidityof hyperbolicright-angledCoxetergroups in
dimension 4and 5
TomoshigeYukita
Contents
Section 1
Section 2
Known Results
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Local rigidityof hyperbolicright-angledCoxetergroups in
dimension 4and 5
TomoshigeYukita
Contents
Section 1
Section 2
Section 2Idea of Proof
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Local rigidityof hyperbolicright-angledCoxetergroups in
dimension 4and 5
TomoshigeYukita
Contents
Section 1
Section 2
Vertical projection
・p : Ud → Rd−1; (x1, · · · , xd−1, t) 7→ (x1, · · · , xd−1):
thevertical projection.
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Local rigidityof hyperbolicright-angledCoxetergroups in
dimension 4and 5
TomoshigeYukita
Contents
Section 1
Section 2
An example
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Local rigidityof hyperbolicright-angledCoxetergroups in
dimension 4and 5
TomoshigeYukita
Contents
Section 1
Section 2
Construction of the 5-polytope P
Consider the regular Euclidean hypercube C in R4.
Take 48 hyperplanes in U5 as follows:・8 hyperplanes
corresponding to the facets of C .・8 hemispheres of radius 1
centered at the centroids of facetsof C .
・32 hemispheres of radius 1 centered at the middle points
ofedges of C .
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Local rigidityof hyperbolicright-angledCoxetergroups in
dimension 4and 5
TomoshigeYukita
Contents
Section 1
Section 2
Verify the combinatorics of P by CoxIter
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Local rigidityof hyperbolicright-angledCoxetergroups in
dimension 4and 5
TomoshigeYukita
Contents
Section 1
Section 2
Rigidity of the shapes of links of ideal vertices.
・Ãd∞: reflection group associated with Euclidean d-cube.・By
Poincaré extension, Ãd∞ < Gd : discrete subgroup.・ρ0 : Ãd∞ →
Gd : the inclusion map.Lemma
・d = 3, 4. ・ρ : Ãd∞ → Isom(U5): representation near by ρ0.Then,
ρ is faithful discrete representation having unique fixedpoint in
boundary at infinity.
・The Lemma implies that for any representation ρ of Ãd∞ nearby
ρ0, the mutual position of hyperplane does not change.
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Local rigidityof hyperbolicright-angledCoxetergroups in
dimension 4and 5
TomoshigeYukita
Contents
Section 1
Section 2
Local rigidity of ΓP
・Γ: RACG with totally geodesic boundaries obtained from ΓP .⇝
ideal vertex subgroups of Γ are isomorphic to Ãd∞ (d = 3, 4).
・ρ ∈ R(Γ, Isom(U5)): a representation near by ρ0.⇝ for any ideal
vertices v , ρv : Ãd∞ → Isom(U5).⇝ the mutual position of
hyperplane of Γ does not change.This fact implies that ΓP is
locally rigid.
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Local rigidityof hyperbolicright-angledCoxetergroups in
dimension 4and 5
TomoshigeYukita
Contents
Section 1
Section 2
Kerckhoff-Storm conjecture
Theorem (S.P.Kerckhoff-P.A.Storm, 2012)
For d ≥ 4, holonomy representations of compact
hyperbolicd-manifolds with totally geodesic boundaries are locally
rigid.
On the other hand, by using coloring technique
byKolpakov-Slavich(2016), we can construct hyperbolic4-manifolds of
finite-volume with totally geodesic boundariesthat are not locally
rigid.
Conjecture (S.P.Kerckhoff-P.A.Storm, 2012)
For d ≥ 5, holonomy representations of hyperbolicd-manifolds of
finite-volume with totally geodesic boundariesare locally
rigid.
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Section 1Section 2