zbMATH — the first resource for mathematics

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.
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
Full Text: DOI arXiv
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.