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Hyperbolic 3-manifolds with k-free fundamental group

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posted on 15.04.2014 by Rosemary K. Guzman
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).

History

Advisor

Shalen, Peter B.

Department

Mathematics

Degree Grantor

University of Illinois at Chicago

Degree Level

Doctoral

Committee Member

Culler, Marc Shipley, Brooke Canary, Richard D. Farb, Benson

Submitted date

2011-08

Language

en

Issue date

07/12/2012

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