The infinitesimal projective rigidity under Dehn filling

TL;DR

This paper investigates the conditions under which Dehn fillings of hyperbolic manifolds are projectively rigid, establishing a link between infinitesimal rigidity of the canonical structure and the rigidity of infinitely many fillings, with explicit examples.

Contribution

It proves that infinitesimal projective rigidity relative to the boundary implies infinite families of projectively rigid Dehn fillings, and provides explicit examples for specific manifolds.

Findings

Infinitesimal rigidity implies infinite rigid Dehn fillings.

Explicit families of rigid fillings for the figure eight knot and Whitehead link.

Connection between boundary rigidity and filling rigidity.

Abstract

To a hyperbolic manifold one can associate a canonical projective structure and ask whether it can be deformed or not. In a cusped manifold, one can ask about the existence of deformations that are trivial on the boundary. We prove that if the canonical projective structure of a cusped manifold is infinitesimally projectively rigid relative to the boundary, then infinitely many Dehn fillings are projectively rigid. We analyze in more detail the figure eight knot and the Withehead link exteriors, for which we can give explicit infinite families of slopes with projectively rigid Dehn fillings.