#### Volume 21, issue 3 (2021)

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

### Sylvain Bonnot, André de Carvalho, Juan González-Meneses and Toby Hall

Algebraic & Geometric Topology 21 (2021) 1351–1370
##### Abstract

There are two objects naturally associated with a braid $\beta \in {B}_{n}$ of pseudo-Anosov type: a (relative) pseudo-Anosov homeomorphism ${\phi }_{\beta }:{S}^{2}\to {S}^{2}$; 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 ${S}^{3}\phantom{\rule{-0.17em}{0ex}}$. We show the disconnect between these objects, by exhibiting a family of braids $\left\{{\beta }_{q}:q\in ℚ\cap \left(0,\frac{1}{3}\right]\right\}$ with the properties that, on the one hand, there is a fixed homeomorphism ${\phi }_{0}:{S}^{2}\to {S}^{2}$ to which the (suitably normalized) homeomorphisms ${\phi }_{{\beta }_{q}}$ 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 $\underset{k\to \infty }{lim}{M}_{{\beta }_{{q}_{k}}}\phantom{\rule{-0.17em}{0ex}}$, for sequences ${q}_{k}\to 0$.

##### Keywords
hyperbolic $3$–manifolds, pseudo-Anosov homeomorphisms, geometric limits
##### Mathematical Subject Classification 2010
Primary: 57M50
Secondary: 20F36, 37E30, 57M25