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

## TL;DR

This paper demonstrates a disconnect between pseudo-Anosov homeomorphisms and hyperbolic 3-manifolds by constructing a family of braids where the associated homeomorphisms converge but the hyperbolic structures diverge in the geometric limit.

## Contribution

It constructs explicit examples of pseudo-Anosov braids showing convergence of homeomorphisms but divergence of hyperbolic 3-manifolds in the geometric limit.

## Key findings

- Homeomorphisms converge to a fixed map as parameters approach zero.
- Hyperbolic 3-manifolds form infinitely many distinct geometric limits.
- The disconnect illustrates different behaviors of surface homeomorphisms and 3-manifold geometries.

## Abstract

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 $q_k\to 0$.