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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∪
Sn-1∞. The restriction to
S1∞ maps onto S2∞
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.
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