Item

Abstract

We show that any closed hyperbolic 3-manifold MMM has a co-final tower of covers Mi→MM_i \to MMi​→M of degrees nin_ini​ such that any subgroup of π1(Mi)\pi_1(M_i)π1​(Mi​) generated by kik_iki​ elements is free, where ki≥niCk_i \ge n_i^Cki​≥niC​ and C=C(M)>0C = C(M) > 0C=C(M)>0. Together with this result we prove that log⁡ki≥C1\sys1(Mi)\log k_i \ge C_1 \sys_1(M_i)logki​≥C1​\sys1​(Mi​), where \sys1(Mi)\sys_1(M_i)\sys1​(Mi​) denotes the systole of MiM_iMi​, thus providing a large set of new examples for a conjecture of Gromov. In the second theorem C1>0C_1 > 0C1​>0 is an absolute constant. We also consider a generalization of these results to non-compact finite volume hyperbolic 3-manifolds.