## Geometry & Topology

### Group invariant Peano curves

#### 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 $Bn=Hn∪S∞n−1$. The restriction to $S∞1$ maps onto $S∞2$ 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
Revised: 12 April 2007
Accepted: 12 April 2007
First available in Project Euclid: 20 December 2017

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

#### 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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