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Hyperbolic 3-Manifolds and Forcing of Surface Automorphisms

Grant number: 12/13615-0
Support type:Scholarships in Brazil - Post-Doctorate
Effective date (Start): December 01, 2012
Effective date (End): November 30, 2014
Field of knowledge:Physical Sciences and Mathematics - Mathematics - Geometry and Topology
Principal researcher:André Salles de Carvalho
Grantee:Rupert William Venzke
Home Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
Associated research grant:11/16265-8 - Low dimensional dynamics, AP.TEM

Abstract

We study a partial order on surface automorphisms called the "forcing order" that measures relative complexity of maps from a dynamical perspective. While the order can be difficult to compute directly when pseudo-Anosov growth is large, our previous work has succeeded in identifying the low growth braids. Using known results/conjectures about this order for Horseshoe braids, we would like to extend our understanding to the more general setting (working with low growth braids as a test case). The partial order extends to a partial order on (fibered) 3-manifolds and gives insight into the structure of invariants for hyperbolic 3-manifolds, such as volumes and homologies. Our hope is to additionally develop nice formulas for these invariants relative to standard models for realizing 3-manifolds, with the intention of classifying non-trivial families of 3-manifolds.