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Group invariant Peano curves. (English) Zbl 1136.57009
Let $$S$$ be a hyperbolic surface, whose universal cover is the hyperbolic plane $$\mathbb{H}^2$$. A discrete faithful representation of the fundamental group of $$S$$ in the group $$\text{Isom}(\mathbb{H}^3)$$ (or the image of such a representation) is called doubly degenerate if the limit set of the induced group action on the compactification $$\mathbb{H}^3\cup S^2_{\infty}$$ is equal to the sphere $$S^2_{\infty}$$.
One of the aims of this paper is to describe some doubly degenerate groups. The main result is that if $$M$$ is a closed hyperbolic $$3$$-manifold which fibers over the circle with pseudo-Anosov monodromy, then the lift of the inclusion map of the fiber $$S$$ in $$M$$ to the hyperbolic universal covers extends continuously to a map between the compactifications of the covering spaces, and induces at the boundary an equivariant $$S^2_{\infty}$$-filling Peano curve. In this situation, $$S$$ is a closed surface, and the authors conjecture that the result extends to the case where $$S$$ is a once-punctured hyperbolic surface. Evidence for this conjecture is provided by the case of a figure-eight knot complement, which the authors analyze in detail.
The study of sphere-filling curves is based on a theorem by R. L. Moore which gives a condition under which the quotient of the $$2$$-sphere by an equivalence relation induced by a cellular decomposition is homeomorphic to the $$2$$-sphere. In the main example considered, the $$2$$-sphere decomposition is obtained by collapsing two laminations.
The paper under review contains several fundamental ideas and techniques of $$3$$-dimensional geometry and topology, and it has been circulated as a preprint for several years.

##### MSC:
 57M60 Group actions on manifolds and cell complexes in low dimensions 57M50 General geometric structures on low-dimensional manifolds 57N05 Topology of the Euclidean $$2$$-space, $$2$$-manifolds (MSC2010) 57N60 Cellularity in topological manifolds 20F65 Geometric group theory
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