The geometry of k-free hyperbolic 3-manifolds

Abstract

We investigate the geometry of closed, orientable, hyperbolic 3-manifolds whose fundamental groups are k-free for a given integer k ≥ 3. We show that any such manifold M contains a point P with the following property: If S is the set of elements of π 1 (M, P ) represented by loops of length < log(2k−1), then for every subset T ⊂ S, we have rank T ≤ k − 3. This generalizes to all k ≥ 3 results proved in [6] and [10], which have been used to relate the volume of a hyperbolic manifold to its topological properties, and it strictly improves on the result obtained in [11] for k = 5. The proof avoids the use of results about ranks of joins and intersections in free groups that were used in [10] and [11].

Realizable ranks of joins and intersections of subgroups in free groups

“…Below we will show that the GTC is false for all values m ≥ 6, but holds true for m = 4. It must be noted that, very recently, Guzman and Shalen [14] proved the Geometric Conjecture in full generality, without dependence on the Group-Theoretic Conjecture.…”

“…It must be noted that, very recently, Guzman and Shalen [14] proved the Geometric Conjecture in full generality, without dependence on the Group-Theoretic Conjecture.…”