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 $<\log (2k-1)$, then for every subset $T\subset S$, we have $\mathrm{rank}T\le k-3$. This generalizes to all $k\ge 3$ 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].

16 pages