Limits of sequences of pseudo-Anosov maps and of hyperbolic 3-manifolds

There are two objects naturally associated with a braid β∈Bn of pseudo-Anosov type: a (relative) pseudo-Anosov homeomorphism φβ:S2→S2; and the finite-volume complete hyperbolic structure on the 3–manifold Mβ obtained by excising the braid closure of β, together with its braid axis, from S3. We show the disconnect between these objects, by exhibiting a family of braids {βq:q∈ℚ∩(0,13]} with the properties that, on the one hand, there is a fixed homeomorphism φ0:S2→S2 to which the (suitably normalized) homeomorphisms φβq converge as q→0, while, on the other hand, there are infinitely many distinct hyperbolic 3–manifolds which arise as geometric limits of the form limk→∞Mβqk, for sequences qk→0.