Free subgroups of $3$-manifold groups

Abstract

We show that any closed hyperbolic $3$-manifold $M$ has a co-final tower of covers $M_i\to M$ of degrees $n_i$ such that any subgroup of ${\pi }_1(M_i)$ generated by $k_i$ elements is free, where $k_i\ge n_i^C$ and $C=C(M)>0$. Together with this result we show that $\log k_i\ge C_{1}sy{s}_1(M_i)$, where $sy{s}_1(M_i)$ denotes the systole of $M_i$, thus providing a large set of new examples for a conjecture of Gromov. In the second theorem $C_1>0$ is an absolute constant. We also consider a generalization of these results to non-compact finite volume hyperbolic $3$-manifolds.

11 pages; v2: filled a gap in the proof of Theorem 3; v3: includes small corrections to the published version