Geometry & Topology

Group invariant Peano curves

James W Cannon and William P Thurston

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Our main theorem is that, if M is a closed hyperbolic 3–manifold which fibres over the circle with hyperbolic fibre S and pseudo-Anosov monodromy, then the lift of the inclusion of S in M to universal covers extends to a continuous map of B2 to B3, where Bn=HnSn1. The restriction to S1 maps onto S2 and gives an example of an equivariant S2–filling Peano curve. After proving the main theorem, we discuss the case of the figure-eight knot complement, which provides evidence for the conjecture that the theorem extends to the case when S is a once-punctured hyperbolic surface.

Article information

Geom. Topol., Volume 11, Number 3 (2007), 1315-1355.

Received: 12 August 1999
Revised: 12 April 2007
Accepted: 12 April 2007
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx]
Secondary: 57M50: Geometric structures on low-dimensional manifolds 57M60: Group actions in low dimensions 57N05: Topology of $E^2$ , 2-manifolds 57N60: Cellularity

Peano curve group invariance hyperbolic structure 3–manifold pseudo-Anosov diffeomorphism fiber bundle over $S^1$


Cannon, James W; Thurston, William P. Group invariant Peano curves. Geom. Topol. 11 (2007), no. 3, 1315--1355. doi:10.2140/gt.2007.11.1315.

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  • W Abikoff, Two theorems on totally degenerate Kleinian groups, Amer. J. Math. 98 (1976) 109–118
  • F Bonahon, Bouts des variétés hyperboliques de dimension $3$, Ann. of Math. $(2)$ 124 (1986) 71–158
  • J W Cannon, The combinatorial structure of cocompact discrete hyperbolic groups, Geom. Dedicata 16 (1984) 123–148
  • A Fathi, F Laudenbach, V Poenaru, et al, Travaux de Thurston sur les surfaces, Séminaire Orsay, Astérisque 66–67, Société Mathématique de France, Paris (1979) French with an English summary
  • R L Moore, Concerning upper semi-continuous collections of continua, Trans. Amer. Math. Soc. 27 (1925) 416–428
  • D Sullivan, Travaux de Thurston sur les groupes quasi-fuchsiens et les variétés hyperboliques de dimension $3$ fibrées sur $S\sp{1}$, from: “Séminaire Bourbaki 554 (1980)”, Lecture Notes in Math. 842, Springer, Berlin (1981) 196–214
  • W P Thurston, Hyperbolic Structures on 3-manifolds, II: Surface groups and 3–manifolds which fiber over the circle, unpublished
  • W P Thurston, Three-dimensional geometry and topology. Vol. 1, Princeton Mathematical Series 35, Princeton University Press, Princeton, NJ (1997) Edited by Silvio Levy