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On hyperbolic once-punctured-torus bundles. II: fractal tessellations of the plane. (English) Zbl 1119.57007
For a closed hyperbolic surface bundle over the circle, the universal cover of the fiber lifts to an imbedding in hyperbolic \(3\)-space. In a highly influential 1989 preprint, now published [J. Cannon and W. Thurston, Group invariant Peano curves, Geometry and Topology 11, 1315-1355 (2007)], the first author and Thurston showed that this imbedding extends to an equivariant surjective map from the circle at infinity to the sphere at infinity. Various extensions to the nonclosed case have been obtained. The paper under review continues a deep examination, by several authors, of the case when the surface is a once-punctured torus. Its main results give descriptive information about the resulting space-filling curves, in particular showing how they determine certain fractal tesselations of the plane. The authors actually work with a class of space-filling curves obtained from a more general context. From work of C. T. McMullen [Invent. Math. 146, No. 1, 35–91 (2001; Zbl 1061.37025)], W. Thurston [Hyperbolic structures on \(3\)-manifolds II: surface groups and manifolds which fiber over the circle, preprint], F. Bonahon [Ann. Math. (2) 124, 71–158 (1986; Zbl 0671.57008)], Y. N. Minsky [Ann. Math. (2) 149, No. 2, 559–626 (1999; Zbl 0939.30034)], and B. Bowditch [Math. Z. 255, 35–76 (2007)], there is a correspondence between these space-filling curves and irrational bi-infinite sequences in two letters \(R\) and \(L\), where irrational means that the sequence is not eventually constant in either the forward or backward direction. The curves arising from punctured-torus bundles correspond to the periodic irrational sequences. The authors proceed by examining the preimage of a point of the extended complex plane, which may be assumed to be infinity. In the generic and most interesting case, the complement of the preimage is the union of an infinite collection of open arcs. Each arc maps onto a so-called column, which is a string of closed Jordan domains with fractal boundaries, each meeting the next in a single point. These columns fit together to form a fractal tesselation of the plane. The authors work out the combinatorics of the tesselation very precisely in terms of the defining sequence in \(R\) and \(L\), and also determine the symmetry groups of the tesselations.

MSC:
57M50 General geometric structures on low-dimensional manifolds
20E05 Free nonabelian groups
20F28 Automorphism groups of groups
57M05 Fundamental group, presentations, free differential calculus
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