## Geometry & Topology

### Universal circles for quasigeodesic flows

Danny Calegari

#### Abstract

We show that if $M$ is a hyperbolic $3$–manifold which admits a quasigeodesic flow, then $π1(M)$ acts faithfully on a universal circle by homeomorphisms, and preserves a pair of invariant laminations of this circle. As a corollary, we show that the Thurston norm can be characterized by quasigeodesic flows, thereby generalizing a theorem of Mosher, and we give the first example of a closed hyperbolic $3$–manifold without a quasigeodesic flow, answering a long-standing question of Thurston.

#### Article information

Source
Geom. Topol., Volume 10, Number 4 (2006), 2271-2298.

Dates
Revised: 10 September 2006
Accepted: 25 October 2006
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513799803

Digital Object Identifier
doi:10.2140/gt.2006.10.2271

Mathematical Reviews number (MathSciNet)
MR2284058

Zentralblatt MATH identifier
1129.57032

#### Citation

Calegari, Danny. Universal circles for quasigeodesic flows. Geom. Topol. 10 (2006), no. 4, 2271--2298. doi:10.2140/gt.2006.10.2271. https://projecteuclid.org/euclid.gt/1513799803

#### References

• D Calegari, The geometry of $\mathbf{R}$-covered foliations, Geom. Topol. 4 (2000) 457–515
• D Calegari, Circular groups, planar groups, and the Euler class, from: “Proceedings of the Casson Fest (Arkansas and Texas, 2003)”, (C Gordon, Y Rieck, editors), Geom. Topol. Monogr. 7 (2004) 431–491
• D Calegari, N M Dunfield, Laminations and groups of homeomorphisms of the circle, Invent. Math. 152 (2003) 149–204
• J Cannon, W Thurston, Group invariant peano curves, preprint (1985)
• S R Fenley, Laminar free hyperbolic 3-manifolds
• S R Fenley, 1259365
• S R Fenley, Foliations, topology and geometry of 3-manifolds: $\bold R$-covered foliations and transverse pseudo-Anosov flows, Comment. Math. Helv. 77 (2002) 415–490
• S Fenley, L Mosher, Quasigeodesic flows in hyperbolic 3-manifolds, Topology 40 (2001) 503–537
• D Gabai, Foliations and the topology of $3$-manifolds, J. Differential Geom. 18 (1983) 445–503
• D Gabai, W H Kazez, The finiteness of the mapping class group for atoroidal $3$-manifolds with genuine laminations, J. Differential Geom. 50 (1998) 123–127
• D Gabai, W H Kazez, Group negative curvature for $3$-manifolds with genuine laminations, Geom. Topol. 2 (1998) 65–77
• D Gabai, U Oertel, 1005607
• M Gromov, Hyperbolic groups, from: “Essays in group theory”, Math. Sci. Res. Inst. Publ. 8, Springer, New York (1987) 75–263
• D Husemoller, Fibre bundles, third edition, Graduate Texts in Mathematics 20, Springer, New York (1994)
• M Kapovich, Hyperbolic manifolds and discrete groups, Progress in Mathematics 183, Birkhäuser, Boston (2001)
• J Milnor, On the existence of a connection with curvature zero, Comment. Math. Helv. 32 (1958) 215–223
• J W Milnor, J D Stasheff, Characteristic classes, Annals of Mathematics Studies 76, Princeton University Press (1974)
• L Mosher, Laminations and flows transverse to finite depth foliations, Part I: Branched surfaces and dynamics, preprint
• L Mosher, Dynamical systems and the homology norm of a $3$-manifold. I. Efficient intersection of surfaces and flows, Duke Math. J. 65 (1992) 449–500
• L Mosher, Dynamical systems and the homology norm of a $3$-manifold. II, Invent. Math. 107 (1992) 243–281
• L Mosher, Examples of quasi-geodesic flows on hyperbolic $3$-manifolds, from: “Topology '90 (Columbus, OH, 1990)”, Ohio State Univ. Math. Res. Inst. Publ. 1, de Gruyter, Berlin (1992) 227–241
• U Oertel, Homology branched surfaces: Thurston's norm on $H\sb 2(M\sp 3)$, from: “Low-dimensional topology and Kleinian groups (Coventry/Durham, 1984)”, London Math. Soc. Lecture Note Ser. 112, Cambridge Univ. Press (1986) 253–272
• E H Spanier, Algebraic topology, McGraw-Hill Book Co., New York (1966)
• W P Thurston, Hyperbolic Structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle
• W P Thurston, Three-manifolds, Foliations and Circles, II, preprint
• W P Thurston, A norm for the homology of $3$-manifolds, Mem. Amer. Math. Soc. 59 (1986) i–vi and 99–130
• R L Wilder, Topology of manifolds, American Mathematical Society Colloquium Publications 32, American Mathematical Society, Providence, R.I. (1979) Reprint of 1963 edition
• J W Wood, 0248873
• A Zeghib, Sur les feuilletages géodésiques continus des variétés hyperboliques, Invent. Math. 114 (1993) 193–206