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Journal Article

#### Free subgroups of 3-manifold groups

##### External Ressource

https://doi.org/10.4171/GGD/542

(Publisher version)

##### Fulltext (public)

arXiv:1803.05868.pdf

(Preprint), 186KB

##### Supplementary Material (public)

There is no public supplementary material available

##### Citation

Belolipetsky, M., & Dória, C. (2020). Free subgroups of 3-manifold groups.* Groups, Geometry, and Dynamics,* *14*(1), 243-254. doi:10.4171/GGD/542.

Cite as: http://hdl.handle.net/21.11116/0000-0006-D82F-3

##### 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.