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 $k\ge 3$. We show that any such manifold $M$ contains a point $P$ of $M$ with the following property: If $S$ is the set of elements of $\pi_1(M,P)$ represented by loops of length $<\log(2k1)$, then for every subset $T \subset S$, we have ${\rm rank}\ T \le k3$. This generalizes to all $k\ge3$ 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].
 Publication:

arXiv eprints
 Pub Date:
 February 2018
 arXiv:
 arXiv:1802.08350
 Bibcode:
 2018arXiv180208350G
 Keywords:

 Mathematics  Geometric Topology;
 57M50
 EPrint:
 16 pages