Free subgroups of $3$-manifold groups

TL;DR

This paper demonstrates that hyperbolic 3-manifold groups have co-final towers where subgroups generated by many elements are free, linking subgroup complexity to geometric properties like systole, and extends results to non-compact cases.

Contribution

It establishes the existence of co-final towers with free subgroups of large rank in hyperbolic 3-manifold groups and relates subgroup size to systole, advancing understanding of 3-manifold group structures.

Findings

Existence of co-final towers with free subgroups of rank ≥ n_i^C

Logarithm of subgroup rank bounds the systole of the manifold

Results extend to non-compact finite volume hyperbolic 3-manifolds

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 $\log k_i \geq C_1 sys_1(M_i)$, 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 non-compact finite volume hyperbolic $3$-manifolds.