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Primitive stable representations of free Kleinian groups. (English) Zbl 1361.57025
Let $$F$$ be a non-abelian free group of rank $$n$$. The authors give a complete criterion for a discrete faithful representation $$\rho : F_{n} \to \text{PSL} (2, \mathbb C)$$ to be primitive stable. Theorem 1.1 states that if $$\rho$$ is a discrete faithful representation of $$F$$ without parabolics then $$\rho$$ is primitive stable. Theorem 1.2 states the following. Let $$\rho$$ be a discrete, faithful and geometrically infinite representation with parabolics such that the non-cuspidal part $$M_{0}$$ of $$M = \mathbb H^{3} / \rho (F)$$ is the union of the relative compact core $$H$$ and finitely many end neighbourhoods $$E_{i}$$ facing $$S_{i} \subset \partial H$$. Then the representation $$\rho$$ is primitive stable if and only if every parabolic curve is disc-busting, and every geometrically infinite end $$E_{i}$$ has the ending lamination $$\lambda_{i}$$ which is disc-busting on $$\partial H$$. These results answer Y. N. Minsky’s conjectures [Isr. J. Math. 193, 47–70 (2013; Zbl 1282.57023)] about geometric conditions on $$\mathbb H^{3} / \rho(F_{n})$$ regarding the primitive stability of $$\rho$$.

##### MSC:
 57M60 Group actions on manifolds and cell complexes in low dimensions 20F67 Hyperbolic groups and nonpositively curved groups 20E05 Free nonabelian groups
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