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 $B^2$ 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.