## Geometry & Topology

### Discrete primitive-stable representations with large rank surplus

#### Abstract

We construct a sequence of primitive-stable representations of free groups into $PSL2(ℂ)$ 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.

#### Article information

Source
Geom. Topol., Volume 17, Number 4 (2013), 2223-2261.

Dates
Received: 30 September 2010
Revised: 16 October 2012
Accepted: 25 April 2013
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513732651

Digital Object Identifier
doi:10.2140/gt.2013.17.2223

Mathematical Reviews number (MathSciNet)
MR3109867

Zentralblatt MATH identifier
1278.57026

#### Citation

Minsky, Yair N; Moriah, Yoav. Discrete primitive-stable representations with large rank surplus. Geom. Topol. 17 (2013), no. 4, 2223--2261. doi:10.2140/gt.2013.17.2223. https://projecteuclid.org/euclid.gt/1513732651

#### References

• I Agol, Tameness of hyperbolic $3$–manifolds
• F Bonahon, Bouts des variétés hyperboliques de dimension $3$, Ann. of Math. 124 (1986) 71–158
• F Bonahon, Geometric structures on $3$–manifolds, from: “Handbook of geometric topology”, (R J Daverman, R B Sher, editors), North-Holland, Amsterdam (2002) 93–164
• D Calegari, D Gabai, Shrinkwrapping and the taming of hyperbolic $3$–manifolds, J. Amer. Math. Soc. 19 (2006) 385–446
• R D Canary, A covering theorem for hyperbolic $3$–manifolds and its applications, Topology 35 (1996) 751–778
• R D Canary, C J Leininger, Kleinian groups with discrete length spectrum, Bull. Lond. Math. Soc. 39 (2007) 189–193
• J Hempel, $3$–manifolds, Ann. of Math. Studies 86, Princeton Univ. Press (1976)
• W H Jaco, Lectures on three-manifold topology, CBMS Regional Conference Series in Mathematics 43, Amer. Math. Soc. (1980)
• W H Jaco, P B Shalen, Seifert fibered spaces in $3$–manifolds, Mem. Amer. Math. Soc. 21 (1979) viii+192
• K Johannson, Homotopy equivalences of $3$–manifolds with boundaries, Lecture Notes in Mathematics 761, Springer, Berlin (1979)
• R S Kulkarni, P B Shalen, On Ahlfors' finiteness theorem, Adv. Math. 76 (1989) 155–169
• A Lubotzky, Dynamics of $\mathrm{Aut}(F_n)$ actions on group presentations and representations, from: “Geometry, rigidity, and group actions”, Univ. Chicago Press (2011) 609–643
• M Lustig, Y Moriah, $3$–manifolds with irreducible Heegaard splittings of high genus, Topology 39 (2000) 589–618
• D McCullough, Compact submanifolds of $3$–manifolds with boundary, Quart. J. Math. Oxford Ser. 37 (1986) 299–307
• Y N Minsky, On dynamics of $\mathrm{Out}(F\sb n)$ on ${\rm PSL}\sb 2({\Bbb{C}})$ characters, Israel J. Math. 193 (2013) 47–70
• Y Moriah, J Schultens, Irreducible Heegaard splittings of Seifert fibered spaces are either vertical or horizontal, Topology 37 (1998) 1089–1112
• G P Scott, Compact submanifolds of $3$–manifolds, J. London Math. Soc. 7 (1973) 246–250
• W Thurston, Hyperbolic geometry and $3$–manifolds, from: “Low-dimensional topology”, (R Brown, T L Thickstun, editors), London Math. Soc. Lecture Note Ser. 48, Cambridge Univ. Press (1982) 9–25
• J H C Whitehead, On certain sets of elements in a free group, Proc. London Math. Soc. S2-41 (1936) 48
• J H C Whitehead, On equivalent sets of elements in a free group, Ann. of Math. 37 (1936) 782–800