A spherical CR structure on the complement of the figure eight knot with discrete holonomy

Falbel, Elisha

doi:10.4310/jdg/1207834658

Cited by 48 publications

(102 citation statements)

References 8 publications

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“…(Examples of discrete representations of certain hyperbolic 3-manifold groups into PU.2; 1/ are already known, see Falbel [9] and Schwartz [22]. )…”

Section: Introductionmentioning

confidence: 99%

Flexing closed hyperbolic manifolds

2007

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We show that for certain closed hyperbolic manifolds, one can nontrivially deform the real hyperbolic structure when it is considered as a real projective structure. It is also shown that in the presence of a mild smoothness hypothesis, the existence of such real projective deformations is equivalent to the question of whether one can nontrivially deform the canonical representation of the real hyperbolic structure when it is considered as a group of complex hyperbolic isometries. The set of closed hyperbolic manifolds for which one can do this seems mysterious. 57M50

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“…(Examples of discrete representations of certain hyperbolic 3-manifold groups into PU.2; 1/ are already known, see Falbel [9] and Schwartz [22]. )…”

Section: Introductionmentioning

confidence: 99%

Flexing closed hyperbolic manifolds

2007

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“…Note that this definition is related to the definition of a symmetric tetrahedron introduced in [F2,Section 4.3]. There, a configuration [p 0 , p 1 , p 2 , p 3 ] is symmetric if there exists an antiholomorphic involution ϕ such that ϕ(p i ) = p j and ϕ(p k ) = p l for {i, j, k, l} = {0, 1, 2, 3}.…”

Section: A P(c)-valued Cr Invariantmentioning

confidence: 99%

A combinatorial invariant for spherical CR structures

Falbel

Wang

2013

Asian Journal of Mathematics

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We study a cross-ratio of four generic points of S 3 which comes from spherical CR geometry. We construct a homomorphism from a certain group generated by generic configurations of four points in S 3 to the pre-Bloch group P(C). If M is a 3-dimensional spherical CR manifold with a CR triangulation, by our homomorphism, we get a P(C)-valued invariant for M . We show that when applying to it the Bloch-Wigner function, it is zero. Under some conditions on M , we show the invariant lies in the Bloch group B(k), where k is the field generated by the cross-ratio. For a CR triangulation of Whitehead link complement, we show its invariant is a torsion in B(k) and for a triangulation of the complement of the 5 2 -knot we show that the invariant is not trivial and not a torsion element.

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“…Twisted S 1 -bundles admit many such structures (see for example [88]), but recently Ananin, Grossi and Gusevskii [4,5] have constructed surprising examples of spherical CR-structures on products of closed hyperbolic surfaces with S 1 . Other interesting examples of spherical CR-structures on 3-manifolds have been constructed by Schwartz [144,145,146], Falbel [57], Gusevskii, Parker [137], Parker-Platis [138].…”

Section: Complex Projective 1-manifolds Flat Conformal Structures Anmentioning

confidence: 99%

Locally Homogeneous Geometric Manifolds

Goldman

2011

Proceedings of the International Congress of Mathematicians 2010 (ICM 2010)

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Abstract.Motivated by Felix Klein's notion that geometry is governed by its group of symmetry transformations, Charles Ehresmann initiated the study of geometric structures on topological spaces locally modeled on a homogeneous space of a Lie group. These locally homogeneous spaces later formed the context of Thurston's 3-dimensional geometrization program. The basic problem is for a given topology Σ and a geometry X = G/H, to classify all the possible ways of introducing the local geometry of X into Σ. For example, a sphere admits no local Euclidean geometry: there is no metrically accurate Euclidean atlas of the earth. One develops a space whose points are equivalence classes of geometric structures on Σ, which itself exhibits a rich geometry and symmetries arising from the topological symmetries of Σ.We survey several examples of the classification of locally homogeneous geometric structures on manifolds in low dimension, and how it leads to a general study of surface group representations. In particular geometric structures are a useful tool in understanding local and global properties of deformation spaces of representations of fundamental groups.Mathematics Subject Classification (2000). Primary 57M50; Secondary 57N16.Keywords. connection, curvature, fiber bundle, homogeneous space, Thurston geometrization of 3-manifolds, uniformization, crystallographic group, discrete group, proper action, Lie group, fundamental group, holonomy, completeness, development, geodesic, symplectic structure, Teichmüller space, Fricke space, hypebolic structure, Riemannian metric, Riemann surface, affine structure, projective structure, conformal structure, spherical CR structure, complex hyperbolic structure, deformation space, mapping class group, ergodic action. Historical backgroundWhile geometry involves quantitative measurements and rigid metric relations, topology deals with the loose quantitative organization of points. Felix Klein proposed in his 1872 Erlangen Program that the classical geometries be considered as the properties of a space invariant under a transitive Lie group action. Therefore one may ask which topologies support a system of local coordinates modeled on a fixed homogeneous space X = G/H such that on overlapping coordinate patches, the coordinate changes are locally restrictions of transformations from G. W. GoldmanIn this generality this question was first asked by Charles Ehresmann [55] at the conference "Quelques questions de Geométrie et de Topologie," in Geneva in 1935. Forty years later, the subject of such locally homogeneous geometric structures experienced a resurgence when W. Thurston placed his 3-dimensional geometrization program [158] in the context of locally homogeneous (Riemannian) structures. The rich diversity of geometries on homogeneous spaces brings in a wide range of techniques, and the field has thrived through their interaction.Before Ehresmann, the subject may be traced to several independent threads in the 19th century:• The theory of monodromy of Schwarzian differential equ...

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A spherical CR structure on the complement of the figure eight knot with discrete holonomy

Cited by 48 publications

References 8 publications

Flexing closed hyperbolic manifolds

Flexing closed hyperbolic manifolds

A combinatorial invariant for spherical CR structures

Locally Homogeneous Geometric Manifolds

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