Local rigidity of hyperbolic right-angled Coxeter groups in ...han/slide2019/yukita.pdfCoxeter...

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

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Section 2

Contents

1. Definitions and main result

2. Idea of proof

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Section 1Definitions and main result

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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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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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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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An example of hyperbolic Coxeter group

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Main result

Theorem (Y)


Idea



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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.

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Main result

Theorem (Y)


Idea



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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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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 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).

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Main result

Theorem (Y)


Idea



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Known Results

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Section 2Idea of Proof

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Vertical projection

・p : Ud → Rd−1; (x1, · · · , xd−1, t) 7→ (x1, · · · , xd−1): thevertical projection.

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An example

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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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Verify the combinatorics of P by CoxIter

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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 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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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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Local rigidity of hyperbolic right-angled Coxeter groups in ...han/slide2019/yukita.pdfCoxeter...

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