Mega-bimodules of topological polynomials: sub-hyperbolicity and Thurston obstructions
Kelsey, Gregory A.
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https://hdl.handle.net/2142/24082
Description
Title
Mega-bimodules of topological polynomials: sub-hyperbolicity and Thurston obstructions
Author(s)
Kelsey, Gregory A.
Issue Date
2011-05-25T15:02:08Z
Director of Research (if dissertation) or Advisor (if thesis)
Kapovitch, Ilia
Doctoral Committee Chair(s)
Merenkov, Sergiy A.
Committee Member(s)
Kapovitch, Ilia
Leininger, Christopher J.
Athreya, Jayadev S.
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Combinatorics of complex dynamics
self-similar groups
Abstract
In 2006, Bartholdi and Nekrashevych solved a decade-old problem in holomorphic dynamics by creatively applying the theory of self-similar groups. Nekrashevych expanded this work in 2009 to define what we refer to as mega-bimodules which capture the topological data of Hurwitz classes of topological polynomials. He also showed that proving that these mega-bimodules are sub-hyperbolic will have two important implications: that all iterated monodromy groups of topological polynomials are contracting and that the Hubbard- Schliecher spider algorithm for complex polynomials generalizes to topological polynomials. We prove sub- hyperbolicity in the simplest non-trivial case and apply these mega-bimodules to holomorphic dynamics to prove a partial converse to the Berstein-Levy Theorem proved in 1985.
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