#### Volume 11, issue 3 (2007)

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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 ${B}^{2}$ 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