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

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

doi:10.2140/agt.2021.21.1351

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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

DOI: 10.2140/agt.2021.21.1351

Abstract

There are two objects naturally associated with a braid $β \in B_{n}$ of pseudo-Anosov type: a (relative) pseudo-Anosov homeomorphism $φ_{β} : S^{2} \to S^{2}$ ; 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 ${β_{q} : q \in ℚ \cap (0, \frac{1}{3}]}$ with the properties that, on the one hand, there is a fixed homeomorphism $φ_{0} : S^{2} \to S^{2}$ 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 $q_{k} \to 0$ .

Keywords

hyperbolic $3$–manifolds, pseudo-Anosov homeomorphisms, geometric limits

Mathematical Subject Classification 2010

Primary: 57M50

Secondary: 20F36, 37E30, 57M25

References

Publication

Received: 15 February 2019

Revised: 5 December 2019

Accepted: 8 March 2020

Published: 11 August 2021

Authors

Sylvain Bonnot
Instituto de Matemática e Estatística Universidade de São Paulo São Paulo Brazil

André de Carvalho
Instituto de Matemática e Estatística Universidade de São Paulo São Paulo Brazil

Juan González-Meneses
Departamento de Álgebra Instituto de Matemáticas Universidad de Sevilla Sevilla Spain

Toby Hall
Department of Mathematical Sciences University of Liverpool Liverpool United Kingdom

Volume 21, issue 3 (2021)