Resumen de Similar fillings and isolation of cusps of hyperbolic 3-manifolds

Roberto Frigerio

• We deepen the analysis of certain classes Mg,k of hyperbolic 3-manifolds that were introduced in a previous work by B. Martelli, C. Petronio and the author. Each element of Mg,k is an oriented complete finite-volume hyperbolic 3-manifold with compact connected geodesic boundary of genus g and k cusps. We prove that several elements in Mg,k admit nonhomeomorphic hyperbolic Dehn fillings sharing the same volume, homology, cusp volume, cusp shape, Heegaard genus, complex length of the shortest geodesic, length of the shortest return path, and Turaev¿Viro invariants.

  Let N be a complete finite-volume hyperbolic 3-manifold with (possibly empty) geodesic boundary and cusps C1,¿,Ch,Ch+1,¿,Ck. According to Neumann and Reid, the cusps C1,¿,Ch are said to be geometrically isolated from Ch+1,¿,Ck if any small deformation of the hyperbolic structure on N induced by Dehn filling Ch+1,¿,Ck does not affect the Euclidean structure at C1,¿,Ch. We show here that the cusps of any manifold in Mg,k are geometrically isolated from each other. On the contrary, any element in Mg,k admits an infinite family of hyperbolic Dehn fillings inducing nontrivial deformations of the hyperbolic structure on the geodesic boundary.