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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
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