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Discrete primitive-stable representations with large rank surplus. (English) Zbl 1278.57026
In [Isr. J. Math. 193, 47–70 (2013; Zbl 1282.57023)], Y. R. Minsky exams the Out(\(F_n\)) action on the \(PSL_2(\mathbb C)\)-character variety of the free group \(F_n\) and in particular the action on the subsets of discrete faithful representations and of representations with dense image. Minsky showed there is an open set of primitive-stable representations which contains all Shottky (discrete, faithful, convex-cocompact) representations on which Out(\(F_n\)) acts properly discontinuously.
In the paper reviewed here, Minsky and Moriah study representations corresponding to fundamental groups of finite-volume hyperbolic 3-manifolds; that is, representations whose images are discrete, torsion-free subgroups such that the corresponding hyperbolic 3-manifold has finite volume. Note that by Mostow rigidity, the image of the character of such a representation depends uniquely on the isomorphism class of the fundamental group. However such groups have presentations of arbitrarily high rank. In the main theorem, the authors show that there exist infinite sequences of primitive-stable representations \(\rho^r\) of \(F_{n_0+2}\) in \(PSL_2(\mathbb C)\), such that each \(\rho^r\) has discrete, torsion-free image and such that the quotient manifolds \(\mathbb H^3/\rho^r(F_{n_0+2r})\) converge geometrically to a knot complement in \(S^3\).

MSC:
57M60 Group actions on manifolds and cell complexes in low dimensions
57M50 General geometric structures on low-dimensional manifolds
57M05 Fundamental group, presentations, free differential calculus
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