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Minimal index torsion-free subgroups of Kleinian groups. (English) Zbl 0890.57016
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.

##### MSC:
 57M50 General geometric structures on low-dimensional manifolds 30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
Zbl 0498.20033
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