Free subgroups of $3$manifold groups
Abstract
We show that any closed hyperbolic $3$manifold $M$ has a cofinal 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 $\log k_i \geq C_1 sys_1(M_i)$, where $sys_1(M_i)$ 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 noncompact finite volume hyperbolic $3$manifolds.
 Publication:

arXiv eprints
 Pub Date:
 March 2018
 arXiv:
 arXiv:1803.05868
 Bibcode:
 2018arXiv180305868B
 Keywords:

 Mathematics  Group Theory;
 Mathematics  Geometric Topology;
 30F40 (Primary);
 53C23;
 57M50 (Secondary)
 EPrint:
 11 pages