Discrete primitive-stable representations with large rank surplus

Minsky, Yair N¹ ; Moriah, Yoav²

¹ Department of Mathematics, Yale University, PO Box 208283, New Haven, CT 06520, USA
² Department of Mathematics, Technion, 32000 Haifa, Israel

Geometry & topology, Tome 17 (2013) no. 4, pp. 2223-2261.

Résumé

We construct a sequence of primitive-stable representations of free groups into ${PSL}_2\left(ℂ\right)$ whose ranks go to infinity, but whose images are discrete with quotient manifolds that converge geometrically to a knot complement. In particular this implies that the rank and geometry of the image of a primitive-stable representation imposes no constraint on the rank of the domain.

Détail

DOI : 10.2140/gt.2013.17.2223

Classification : 57M60, 57M50, 57M05
Keywords: primitive stable, Whitehead graph, representation, rank, Dehn filling

Bibliographie

[1] I Agol, Tameness of hyperbolic $3$–manifolds

[2] F Bonahon, Bouts des variétés hyperboliques de dimension $3$, Ann. of Math. 124 (1986) 71

[3] F Bonahon, Geometric structures on $3$–manifolds, from: "Handbook of geometric topology" (editors R J Daverman, R B Sher), North-Holland (2002) 93

[4] D Calegari, D Gabai, Shrinkwrapping and the taming of hyperbolic $3$–manifolds, J. Amer. Math. Soc. 19 (2006) 385

[5] R D Canary, A covering theorem for hyperbolic $3$–manifolds and its applications, Topology 35 (1996) 751

[6] R D Canary, C J Leininger, Kleinian groups with discrete length spectrum, Bull. Lond. Math. Soc. 39 (2007) 189

[7] J Hempel, $3$–manifolds, Ann. of Math. Studies 86, Princeton Univ. Press (1976)

[8] W H Jaco, Lectures on three-manifold topology, CBMS Regional Conference Series in Mathematics 43, Amer. Math. Soc. (1980)

[9] W H Jaco, P B Shalen, Seifert fibered spaces in $3$–manifolds, Mem. Amer. Math. Soc. 21 (1979)

[10] K Johannson, Homotopy equivalences of $3$–manifolds with boundaries, Lecture Notes in Mathematics 761, Springer (1979)

[11] R S Kulkarni, P B Shalen, On Ahlfors' finiteness theorem, Adv. Math. 76 (1989) 155

[12] A Lubotzky, Dynamics of $\mathrm{Aut}(F_n)$ actions on group presentations and representations, from: "Geometry, rigidity, and group actions", Univ. Chicago Press (2011) 609

[13] M Lustig, Y Moriah, $3$–manifolds with irreducible Heegaard splittings of high genus, Topology 39 (2000) 589

[14] D Mccullough, Compact submanifolds of $3$–manifolds with boundary, Quart. J. Math. Oxford Ser. 37 (1986) 299

[15] Y N Minsky, On dynamics of $\mathrm{Out}(F_n)$ on $\mathrm{PSL}_2({\mathbb{C}})$ characters, Israel J. Math. 193 (2013) 47

[16] Y Moriah, J Schultens, Irreducible Heegaard splittings of Seifert fibered spaces are either vertical or horizontal, Topology 37 (1998) 1089

[17] G P Scott, Compact submanifolds of $3$–manifolds, J. London Math. Soc. 7 (1973) 246

[18] W Thurston, Hyperbolic geometry and $3$–manifolds, from: "Low-dimensional topology" (editors R Brown, T L Thickstun), London Math. Soc. Lecture Note Ser. 48, Cambridge Univ. Press (1982) 9

[19] J H C Whitehead, On certain sets of elements in a free group, Proc. London Math. Soc. S2-41 (1936) 48

[20] J H C Whitehead, On equivalent sets of elements in a free group, Ann. of Math. 37 (1936) 782