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Primitive stable closed hyperbolic 3-manifolds. (English) Zbl 1297.57039
Let $$M$$ be a closed, orientable 3-manifold. Assume that Heegaard surfaces have genus at least 2. A representation $$\rho: F \rightarrow \text{PSL}_{2} {\mathbb C}$$ of a free group is primitive if it has the property that any $$\rho$$-equivariant map from a Cayley graph of $$F$$ to $${\mathbb H}^{3}$$ takes the geodesics defined by primitive elements to uniform quasi-geodesics. Let $$\chi (F_{g})$$ be the character variety of the representation of the free group $$F_{g}$$ of rank $$g$$ to $$\text{PSL}_{2} \mathbb {C}$$. Let $$S_{g}$$ be the subspace of $$\chi (F_{g})$$ consisting of Schottky representations. In the paper under review the authors give a sufficient condition for primitive stability of closed hyperbolic 3-manifolds and present its application. Their main result is the following.
For any positive number $$R,$$ there exists a number $$K$$ depending only on $$R$$ and the genus $$g$$ as follows. For any 3-manifold admitting a genus $$g$$ Heegaard splitting $$M=H^{1} \cup_{\Sigma} H^{2}$$ whose Hempel distance is greater than $$K$$ and which has $$R$$-bounded combinatorics, the manifold $$M$$ is hyperbolic and the representation $$\iota_{*}:\pi_{1}(H^{j}) \rightarrow \pi_{1}(M) \subset \text{PSL}_{2} {\mathbb C}$$ is primitive stable for $$j=1,2$$.
The proof employs the result in Namazi’s thesis on model manifolds for Heegaard splittings with $$R$$-bounded combinatorics and the characterization of primitive stable discrete and faithful representations by the authors. As an application, it is shown that every point in the frontier of $$S_{g}$$ in $$\chi (F_{g})$$ is a limit of a sequence of primitive stable unfaithful discrete representations $$\{\rho_{n}\}$$ such that $${\mathbb H}^{3}/ \rho_{n}(F_{g})$$ is a closed hyperbolic 3-manifold for every $$n$$. To prove this, they show that every maximal cusp is a limit of a sequence of primitive stable unfaithful discrete representations corresponding to closed manifolds.
##### MSC:
 57M50 General geometric structures on low-dimensional manifolds 30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization) 51M10 Hyperbolic and elliptic geometries (general) and generalizations 57S25 Groups acting on specific manifolds
##### Keywords:
hyperbolic 3-manifold; primitive stable
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