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Infinitesimal projective rigidity under Dehn filling

Michael Heusener and Joan Porti

Geometry & Topology 15 (2011) 2017–2071

To a hyperbolic manifold one can associate a canonical projective structure and a fundamental question is whether or not it can be deformed. In particular, the canonical projective structure of a finite volume hyperbolic manifold with cusps might have deformations which are trivial on the cusps.

The aim of this article is to prove that if the canonical projective structure on a cusped hyperbolic manifold M is infinitesimally projectively rigid relative to the cusps, then infinitely many hyperbolic Dehn fillings on M are locally projectively rigid. We analyze in more detail the figure eight knot and the Whitehead link exteriors, for which we can give explicit infinite families of slopes with projectively rigid Dehn fillings.

projective structure, variety of representations, infinitesimal deformation
Mathematical Subject Classification 2000
Primary: 57M50
Secondary: 53A20, 53C15
Received: 8 May 2010
Revised: 10 August 2011
Accepted: 13 September 2011
Published: 23 October 2011
Proposed: Walter Neumann
Seconded: David Gabai, Jean-Pierre Otal
Michael Heusener
Laboratoire de Mathématiques, UMR 6620 - CNRS
Université Blaise Pascal
BP 80026
F-63171 Aubiere
Joan Porti
Departament de Matemàtiques
Universitat Autònoma de Barcelona
08193 Bellaterra