Infinitesimal rigidity of a compact hyperbolic 4–orbifold with totally geodesic boundary

Aougab, Tarik; Storm, Peter A.

doi:10.2140/agt.2009.9.537

Cited by 3 publications

(12 citation statements)

References 8 publications

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“…Hyperbolic d-space. Lorentzian (d + 1)-space R d,1 is a (d + 1)-dimensional real vector space equipped with the Lorentzian inner product (1) x, y…”

Section: Preriminaliesmentioning

confidence: 99%

“…Kerckhoff, Storm, and Aougab studied rigidity of RACGs with Fuchsian ends and showed that the followings: first, Aougab and Strom showed that the RACGs with Fuchsian ends obtained from the compact right-angled 120-cell group are locally rigid [1]. After that, Kerckhoff and Storm generalized the result to convex cocompact ddimensional RACGs with Fuchsian ends for d ≥ 4 [6].…”

Section: The Upper Half-spacementioning

confidence: 99%

“…In order to study the rigidity for such a case, Kerckhoff and Storm introduced the notion of discrete groups with Fuchsian ends [5]. In particular, Kerckhoff, Storm, and Aougab studied the local rigidity of RACGs with Fuchsian ends obtained from the 24-cell group and 120-cell group ( [1,5]).…”

Section: Introductionmentioning

confidence: 99%

“…1 is said to be space-like, time-like, and light-like if x is positive, negative, and zero, respectively. Let us denote by H v the Lorentz orthogonal complement of v, that is,…”

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confidence: 99%

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Locally rigid right-angled Coxeter groups with Fuchsian ends in dimension 5

Yukita¹

2019

Preprint

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“…Hyperbolic d-space. Lorentzian (d + 1)-space R d,1 is a (d + 1)-dimensional real vector space equipped with the Lorentzian inner product (1) x, y…”

Section: Preriminaliesmentioning

confidence: 99%

Section: The Upper Half-spacementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…1 is said to be space-like, time-like, and light-like if x is positive, negative, and zero, respectively. Let us denote by H v the Lorentz orthogonal complement of v, that is,…”

mentioning

confidence: 99%

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Locally rigid right-angled Coxeter groups with Fuchsian ends in dimension 5

Yukita¹

2019

Preprint

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“…This group produces an orbifold quotient of H 4 with totally geodesic boundary in a suitable orbifold sense. Again starting with the hyperbolic 120-cell, Aougab and the second author verified Theorem 1.1 for a specific group generated by reflections in 96 walls of the hyperbolic right-angled 120-cell [1]. These 96 walls were chosen so the complementary 24 walls form a maximal collection of pairwise disjoint walls.…”

mentioning

confidence: 97%

Local rigidity of hyperbolic manifolds with geodesic boundary

Kerckhoff

Storm

2012

Journal of Topology

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Let W be a compact hyperbolic n-manifold with totally geodesic boundary. We prove that if n > 3 then the holonomy representation of π 1 (W ) into the isometry group of hyperbolic n-space is infinitesimally rigid.Starting with a complete hyperbolic manifold X, when is it possible to deform its hyperbolic structure? If X is a finite area surface, its metric always has nontrivial deformations through complete hyperbolic metrics. However, in all higher dimensions, the work of Calabi, Weil, and Garland combine to prove that there are no nontrivial deformations through complete metrics when X has finite volume [3,25,6]. In search of flexibility, one is led to study infinite volume hyperbolic manifolds.In dimension 3 infinite volume hyperbolic manifolds have been studied for many years. A particularly well-understood class of such manifolds are the convex cocompact ones. They are topologically the interior of a compact 3-manifold with boundary consisting of surfaces of genus at least 2. Convex cocompact hyperbolic 3-manifolds have large deformation spaces parametrized by conformal structures on their boundary surfaces inherited from the sphere at infinity. In higher dimensions the situation becomes more mysterious, but one expects to find less flexibility than in dimension 3. This article's research began by looking for a large class of infinite volume hyperbolic manifolds, present in all dimensions, and rigid in dimensions greater than 3. In high dimensions it is difficult to construct interesting infinite volume hyperbolic manifolds. One method is to begin with a closed hyperbolic manifold (often constructed by arithmetic tools) that contains an embedded totally geodesic hypersurface and cut along the hypersurface to obtain a compact hyperbolic manifold W n with totally geodesic boundary. There is a canonical extension of such a structure to a complete, infinite volume hyperbolic metric on an open manifold X n without boundary which is diffeomorphic to the interior of W n . This is the class of manifolds we will study.Associated to any hyperbolic manifold is a representation, called the holonomy representation, of its fundamental group in the group Isom(H n ) of isometries of n-dimensional hyperbolic space. When it is a compact hyperbolic manifold W with totally geodesic boundary, the representation is discrete and faithful. If we denote Date: October 27, 2018.

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Locally Rigid Right-angled Coxeter Groups with Fuchsian Ends in Dimension 5

Yukita¹

2022

Tokyo J. Math.

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Infinitesimal rigidity of a compact hyperbolic 4–orbifold with totally geodesic boundary

Abstract: Kerckhoff and Storm conjectured that compact hyperbolic n-orbifolds with totally geodesic boundary are infinitesimally rigid when n > 3. This paper verifies this conjecture for a specific example based on the 4-dimensional hyperbolic 120-cell.

Cited by 3 publications

References 8 publications

Locally rigid right-angled Coxeter groups with Fuchsian ends in dimension 5

Locally rigid right-angled Coxeter groups with Fuchsian ends in dimension 5

Local rigidity of hyperbolic manifolds with geodesic boundary

Locally Rigid Right-angled Coxeter Groups with Fuchsian Ends in Dimension 5

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