Primitive stable closed hyperbolic 3-manifolds.

*(English)*Zbl 1297.57039Let \(M\) 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.

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.

Reviewer: Shigeyasu Kamiya (Okayama)

##### 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 |

##### References:

[1] | Brock, J., Continuity of Thurston’s length function, Geom. Funct. Anal., 10, 741-797, (2000) · Zbl 0968.57011 |

[2] | Canary, R., A covering theorem for hyperbolic 3-manifolds and its applications, Topology, 35, 751-778, (1996) · Zbl 0863.57010 |

[3] | Canary, R.; Culler, M.; Hersonsky, S.; Shalen, P., Approximation by maximal cusps in boundaries of deformation spaces of Kleinian groups, J. Differ. Geom., 64, 57-109, (2003) · Zbl 1069.57004 |

[4] | Harvey, W., Boundary structure of the modular group, (Kra, I.; Maskit, B., Rieman Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference, Ann. Math. Stud., vol. 97, (1981), Princeton) · Zbl 0461.30036 |

[5] | Hempel, J., 3-manifolds as viewed from the curve complex, Topology, 40, 631-657, (2001) · Zbl 0985.57014 |

[6] | Jeon, W.; Kim, I.; Ohshika, K.; Lecuire, C., Primitive stable representations of free Kleinian groups, Israel J. Math., (2014), in press · Zbl 1361.57025 |

[7] | Kim, I.; Lecuire, C.; Ohshika, K., Convergence of freely decomposable Kleinian groups · Zbl 1339.57024 |

[8] | Lecuire, C., An extension of the masur domain, (Spaces of Kleinian Groups, Lond. Math. Soc. Lect. Note Ser., vol. 329, (2006), Cambridge Univ. Press Cambridge), 49-73 · Zbl 1102.30043 |

[9] | Lubotzky, A., Dynamics of \(F_n\) actions on group presentations and representations, (Geometry, Rigidity and Group Actions, Chicago Lect. Math., (2011), University of Chicago Press Chicago, IL), 609-643 · Zbl 1266.20045 |

[10] | Masur, H.; Minsky, Y., Geometry of the complex of curves, I: hyperbolicity, Invent. Math., 138, 103-149, (1999) · Zbl 0941.32012 |

[11] | Masur, H.; Minsky, Y., Geometry of the complex of curves, II: hierarchical structure, Geom. Funct. Anal., 10, 902-974, (2000) · Zbl 0972.32011 |

[12] | Minsky, Y., Kleinian groups and the complex of curves, Geom. Topol., 4, 117-148, (2000) · Zbl 0953.30027 |

[13] | Minsky, Y., On dynamics of \(\mathit{Out}(F_n)\) on \(\mathit{PSL}_2(\mathbb{C})\) characters, Isr. J. Math., 193, 47-70, (2013) |

[14] | Minsky, Y.; Moriah, Y., Discrete primitive-stable representations with large rank surplus, Geom. Topol., 17, 2223-2261, (2013) · Zbl 1278.57026 |

[15] | Namazi, H., Heegaard splittings and hyperbolic geometry, (2005), Stony Brook University, Thesis |

[16] | Namazi, H.; Souto, J., Non-realizability and ending laminations: proof of the density conjecture, Acta Math., 209, 2, 323-395, (2012) · Zbl 1258.57010 |

[17] | Ohshika, K., Realising end invariants by limits of minimally parabolic, geometrically finite groups, Geom. Topol., 15, 827-890, (2011) · Zbl 1241.30014 |

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