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

Abstract

There are two objects naturally associated with a braid $β\in B_n$ of pseudo-Anosov type: a (relative) pseudo-Anosov homeomorphism $φ_β\colon S^2\to S^2$; and the finite volume complete hyperbolic structure on the 3-manifold $M_β$ obtained by excising the braid closure of $β$, together with its braid axis, from $S^3$. We show the disconnect between these objects, by exhibiting a family of braids $\{β_q:q\in{\mathbb{Q}}\cap(0,1/3]\}$ with the properties that: on the one hand, there is a fixed homeomorphism $φ_0\colon S^2\to S^2$ to which the (suitably normalized) homeomorphisms $φ_{β_{q}}$ converge as $q\to 0$; while on the other hand, there are infinitely many distinct hyperbolic 3-manifolds which arise as geometric limits of the form $\lim_{k\to\infty} M_{β_{q_k}}$, for sequences $q_k\to 0$.