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Tameness persists in weakly type-preserving strong limits. (English) Zbl 1058.57010
It is conjectured that any hyperbolic 3-manifold $$M$$ with finitely generated fundamental group is tame (i.e., the interior of a compact manifold), and also the algebraic limit of a sequence of geometrically finite (hence tame) hyperbolic 3-manifolds. In this context, it is proved in the present paper that if a weakly type-preserving sequence of tame hyperbolic 3-manifolds converges strongly (i.e., algebraically and also geometrically in the Gromov-Hausdorff sense), then the limit is tame. From this it is deduced, under the hypotheses that the fundamental group is not a non-trivial free product of surface groups and cyclic groups, that strong convergence may be replaced by algebraic convergence, and also that tame groups are dense in the boundary of the quasiconformal deformation space of a finitely generated geometrically finite 3-manifold (Kleinian group). This generalizes results of R. D. Canary and Y. N. Minsky [J. Differ. Geom. 43, No. 1, 1–41 (1996; Zbl 0856.57011)] to the cusped case, following the same basic framework of the proof but using relative compact cores.

MSC:
 57M50 General geometric structures on low-dimensional manifolds 30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization) 57N10 Topology of general $$3$$-manifolds (MSC2010)
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