Hyperbolic 3-manifolds with k-free fundamental group
Guzman, Rosemary K.
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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).
actions without inversions on a tree
rank-3 subgroups of a free group