Volume 11, issue 3 (2007)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Group invariant Peano curves

James W Cannon and William P Thurston

Geometry & Topology 11 (2007) 1315–1355
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 = Hn Sn1. 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.

Keywords
Peano curve, group invariance, hyperbolic structure, 3–manifold, pseudo-Anosov diffeomorphism, fiber bundle over $S^1$
Mathematical Subject Classification 2000
Primary: 20F65
Secondary: 57M50, 57M60, 57N05, 57N60
References
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Publication
Received: 12 August 1999
Revised: 12 April 2007
Accepted: 12 April 2007
Published: 20 July 2007
Proposed: Dave Gabai
Seconded: Walter Neumann, Joan Birman
Authors
James W Cannon
279 TMCB
Brigham Young University
Provo, UT 84602
USA
William P Thurston
Department of Mathematics
Cornell University
Ithaca, NY 14853-4201
USA