Group invariant Peano curves

Cannon, James W; Thurston, William P

doi:10.2140/gt.2007.11.1315

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Group invariant Peano curves

James W Cannon and William P Thurston

Geometry & Topology 11 (2007) 1315–1355

DOI: 10.2140/gt.2007.11.1315

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 $B^{2}$ to $B^{3}$ , where $B^{n} = H^{n} \cup S_{\infty}^{n - 1}$ . The restriction to $S_{\infty}^{1}$ maps onto $S_{\infty}^{2}$ and gives an example of an equivariant $S^{2}$ –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

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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

Volume 11, issue 3 (2007)