  There are two objects naturally associated with a braid $\beta\in B_n$ of
pseudo-Anosov type: a (relative) pseudo-Anosov homeomorphism
$\varphi_\beta\colon S^2\to S^2$; and the finite volume complete hyperbolic
structure on the 3-manifold $M_\beta$ obtained by excising the braid closure of
$\beta$, together with its braid axis, from $S^3$. We show the disconnect
between these objects, by exhibiting a family of braids
$\{\beta_q:q\in{\mathbb{Q}}\cap(0,1/3]\}$ with the properties that: on the one
hand, there is a fixed homeomorphism $\varphi_0\colon S^2\to S^2$ to which the
(suitably normalized) homeomorphisms $\varphi_{\beta_{q}}$ 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 $\lim_{k\to\infty}
M_{\beta_{q_k}}$, for sequences $q_k\to 0$.
