hide
Free keywords:
Mathematics, Group Theory, Geometric Topology
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 logki≥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.
