Free subgroups of 3-manifold groups (Q2180202)

The authors call a group \(\Gamma\) \(k\)-\textit{free} if every subgroup generated by \(k\) elements is free; they denote by \(\mathcal{N}_{fr}(\Gamma)\) the maximal \(k\) for which the group is \(k\)-free, and call it the \textit{free rank} of \(\Gamma\) (for example, the free rank of the fundamental group of a closed surface of genus \(g\) is \(2g-1\)). Their main results are then as follows. Given a closed hyperbolic 3-orbifold \(M\), there exists a co-final tower of regular finite-sheeted covers \(M_i \to M\) such that \(\mathcal{N}_{fr}(\pi_1(M_i)) \ge\mathrm{vol}(M_i)^C\), for a constant \(C > 0\) depending only on \(M\). The authors ask whether there exists also a universal such constant \(C\), i.e. a constant which does not depend on the manifold \(M\) (this is indeed the case for arithmetic 3-manifolds, see a paper by \textit{M. Belolipetsky} [Geom. Funct. Anal. 23, No. 3, 813--827 (2013; Zbl 1275.57025)]). Furthermore, any closed hyperbolic 3-orbifold admits a sequence of regular manifold covers \(M_i \to M\) such that \(\mathcal{N}_{fr}(\pi_1(M_i)) \ge (1 + \epsilon)^{\mathrm{sys}_1(M_i)}\) where \(\mathrm{sys}_1(M_i)\) is the length of a shortest closed geodesic in \(M_i\) and \(\epsilon > 0\) is a universal constant now (i.e., does not depend on \(M\)); the same is then shown also for finite volume 3-orbifolds, with an appropriate modification of the free rank \(\mathcal{N}_{fr}(\pi_1(M_i))\) (since the groups may contain copies of \(\mathbb{Z} \times \mathbb{Z}\) now, the authors call a group \(k\)-\textit{semifree} if every subgroup generated by \(k\) elements is a free product of free abelian groups). Such a bound for hyperbolic groups in general was conjectured by \textit{M. Gromov} [Geom. Funct. Anal. 19, No. 3, 743--841 (2009; Zbl 1195.58010)].