Hyperbolic 3-Manifolds and Forcing of Surface Automorphisms

• Grant number: 12/13615-0
• Support Opportunities: 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 Investigator: André Salles de Carvalho
• Grantee: Rupert William Venzke
• Host 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.