It was shown by A. L. Edmonds, J. H. Ewing and R. S. Kulkarni [Invent. Math. 69, 331-346 (1982; Zbl 0498.20033)] that the minimal index of a torsion-free subgroup of a finitely generated Fuchsian group of the first kind \(G\) is bounded above by twice the LCM of the orders of the finite subgroups of \(G\). Here the authors show that no such result is possible for Kleinian groups. Specifically, they exhibit a sequence \(\Gamma_k\) of co-compact Kleinian groups for which the ratio of the minimum index to the LCM is arbitrarily large.
The construction of the \(\Gamma_k\) uses generalized triangle groups and unknotting tunnels of 2-bridge knots. The authors also derive some results of independent interest involving these two constructs.