Free subgroups of $3$-manifold groups
| Autor: | Belolipetsky, Mikhail, Dória, Cayo |
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| Rok vydání: | 2018 |
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| Zdroj: | Groups, Geometry, and Dynamics. 14 (2020), 243-254 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.4171/GGD/542 |
| Popis: | 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. Comment: 11 pages; v2: filled a gap in the proof of Theorem 3; v3: includes small corrections to the published version |
| Databáze: | arXiv |
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