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Hyperbolic 3-manifolds with k-free fundamental group

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Title: Hyperbolic 3-manifolds with k-free fundamental group
Author(s): Guzman, Rosemary K.
Advisor(s): Shalen, Peter B.
Contributor(s): Culler, Marc; Shipley, Brooke; Canary, Richard D.; Farb, Benson
Department / Program: Mathematics
Graduate Major: Mathematics
Degree Granting Institution: University of Illinois at Chicago
Degree: PhD, Doctor of Philosophy
Genre: Doctoral
Subject(s): hyperbolic 3-manifolds k-free 4-free fundamental group actions without inversions on a tree rank-3 subgroups of a free group 5-free
Abstract: The results of Marc Culler and Peter Shalen for 2,3 or 4-free hyperbolic 3-manifolds are contingent on properties specific to and special about rank two subgroups of a free group. Here we determine what construction and algebraic information is required in order to make a geometric statement about M a closed, orientable hyperbolic manifold with k-free fundamental group for any value of k greater than four. Main results are both to show what the formulation of the general statement should be, for which Culler and Shalen’s result is a special case, and that it is true modulo a group-theoretic conjecture. A major result is in the k = 5 case of the geometric statement. Specifically, I show that the required group-theoretic conjecture is in fact true in this case, and so the proposed geometric statement when M is 5-free is indeed a theorem. One can then use the existence of a point and knowledge about π1(M,P) resulting from this theorem to attempt to improve the known lower bound on the volume of M, which is currently 3.44 (Culler, Shalen).
Issue Date: 2012-12-07
Genre: thesis
Rights Information: Copyright 2011 Rosemary K. Guzman
Date Available in INDIGO: 2014-04-15
Date Deposited: 2011-08

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