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 u s ∞n-1. The restriction to ∞s1maps onto s∞2 and gisuves 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.