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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

There are two objects naturally associated with a braid β Bn of pseudo-Anosov type: a (relative) pseudo-Anosov homeomorphism φβ: S2 S2; and the finite-volume complete hyperbolic structure on the 3–manifold Mβ obtained by excising the braid closure of β, together with its braid axis, from S3. We show the disconnect between these objects, by exhibiting a family of braids {βq : q (0, 1 3]} with the properties that, on the one hand, there is a fixed homeomorphism φ0: S2 S2 to which the (suitably normalized) homeomorphisms φβq converge as q 0, while, on the other hand, there are infinitely many distinct hyperbolic 3–manifolds which arise as geometric limits of the form limkMβq k, for sequences qk 0.

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hyperbolic $3$–manifolds, pseudo-Anosov homeomorphisms, geometric limits
Mathematical Subject Classification 2010
Primary: 57M50
Secondary: 20F36, 37E30, 57M25
Received: 15 February 2019
Revised: 5 December 2019
Accepted: 8 March 2020
Published: 11 August 2021
Sylvain Bonnot
Instituto de Matemática e Estatística
Universidade de São Paulo
São Paulo
André de Carvalho
Instituto de Matemática e Estatística
Universidade de São Paulo
São Paulo
Juan González-Meneses
Departamento de Álgebra
Instituto de Matemáticas
Universidad de Sevilla
Toby Hall
Department of Mathematical Sciences
University of Liverpool
United Kingdom