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

Author(s)

Bonnot, Sylvain ^[1] ; de Carvalho, Andre ^[1] ; Gonzalez-Meneses, Juan ^[2] ; Hall, Toby ^[3]

Affiliation

^[1] Univ Sao Paulo, Inst Matemat & Estat, Sao Paulo - Brazil

^[2] Univ Seville, Inst Matemat, Dept Algebra, Seville - Spain

^[3] Univ Liverpool, Dept Math Sci, Liverpool, Merseyside - England

• Document type: Journal article
• Source: Algebraic and Geometric Topology; v. 21, n. 3, p. 1351-1370, 2021.
• Web of Science Citations: 0

Abstract

There are two objects naturally associated with a braid beta is an element of B-n of pseudo-Anosov type: a (relative) pseudo-Anosov homeomorphism phi(beta) : S-2 -> S-2; and the finite-volume complete hyperbolic structure on the 3-manifold M beta obtained by excising the braid closure of fi, together with its braid axis, from S-3. We show the disconnect between these objects, by exhibiting a family of braids [beta(q) ; q is an element of Q boolean AND(0, 1/3]] with the properties that, on the one hand, there is a fixed homeomorphism phi(0) : S-2 -> S-2 to which the (suitably normalized) homeomorphisms phi beta(q) 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 lim(k ->infinity)M(beta qk), for sequences q(k) -> 0 (AU)

• FAPESP's process: 16/25053-8 - Dynamics and geometry in low dimensions
• Grantee: André Salles de Carvalho
• Support Opportunities: Research Projects - Thematic Grants