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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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