Geometry & Topology

Group invariant Peano curves

James W Cannon and William P Thurston

Full-text: Open access

Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513799898

Digital Object Identifier
doi:10.2140/gt.2007.11.1315

Mathematical Reviews number (MathSciNet)
MR2326947

Zentralblatt MATH identifier
1136.57009

Subjects
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

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

Citation

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. https://projecteuclid.org/euclid.gt/1513799898


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References

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