Deformations of noncompact projective manifolds
type: Item
instance of: scholarly article
title: Deformations of noncompact projective manifolds
zbMATH Open document ID: 1309.57014
author: Samuel A. Ballas
published in: Algebraic \& Geometric Topology
publication date: 2014-11-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/1210.8419
reviewed by: Shigeyasu Kamiya
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 57M60
zbMATH DE Number: 6369092
zbMATH Keywords: deformation
zbMATH Keywords: strictly convex projective structure
describes a project that uses: Mathematica
MaRDI profile type: Publication
OpenAlex ID: W3099395928
arXiv ID: 1210.8419
cites work: On the solutions of analytic equations
cites work: Q5703400
cites work: Convexes divisibles III
cites work: Divisible convex sets. IV: Boundary structure in dimension 3
cites work: Geometrization of 3-orbifolds of cyclic type. With an appendix: Limit of hyperbolicity for spherical 3-orbifolds by Michael Heusener and Joan Porti
cites work: Q3708977
cites work: Computing Varieties of Representations of Hyperbolic 3-Manifolds into SL(4, ℝ)
cites work: Flexing closed hyperbolic manifolds.
cites work: On convex projective manifolds and cusps
cites work: A Rigidity Theorem for Discrete Subgroups
cites work: Q3808879
cites work: Incompressible surfaces in 2-bridge knot complements
cites work: Infinitesimal projective rigidity under Dehn filling
cites work: Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery
cites work: Q3814341
cites work: Hyperbolic manifolds and discrete groups
cites work: Deformations of hyperbolic \(3\)-cone-manifolds
cites work: Déformations de connexions localement plates
cites work: Examples of strictly convex projective manifolds of finite volume in arbitrary dimension
cites work: Collisions at infinity in hyperbolic manifolds
cites work: Q4028061
cites work: Remarks on knots with two bridges
cites work: Torsion de Reidemeister pour les variétés hyperboliques
cites work: Foundations of Hyperbolic Manifolds
cites work: Parabolic Representations of Knot Groups, I
cites work: Q4877336
cites work: Remarks on the cohomology of groups
cites work: Q4192669
DOI: 10.2140/AGT.2014.14.2595
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review text: In the paper under review the author discusses the existence of strictly convex flexing deformations for noncompact finite-volume 3-manifolds. The author begins by exploring two-bridge knots and links and obtains the following result:NEWLINENEWLINE Let \(M\) be the complement in \(S^{3}\) of \(4_{1}\) (the figure-eight knot), \(5_{2}\), \(6_{1}\), or \(5_{1}^{2}\) (the Whitehead link). Then \(M\) does not admit strictly convex deformations of its complete hyperbolic structure.NEWLINENEWLINE Using some special properties of amphicheiral knots and their representations, the author shows a complementary result:NEWLINENEWLINE Let \(M\) be the complement of a hyperbolic amphicheiral knot, and suppose that \(M\) is infinitesimally projectively rigid relative to the boundary and the longitude is a rigid slope. Then for sufficiently large \(n\), there is a one-dimensional family of strictly convex deformations of the complete hyperbolic structure on \(M(n/0)\). Here \(M(n/0)\) is the orbifold obtained by surgering a solid torus with longitudinal singular locus of cone angle \(2\pi/n\) along the meridian of \(M\).
description: scientific article; zbMATH DE number 6369092