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.


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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