On hyperbolic once-punctured-torus bundles III: Comparing two tessellations of the complex plane

Details On hyperbolic once-punctured-torus bundles III: Comparing two tessellations of the complex plane On hyperbolic once-punctured-torus bundles III: Comparing two tessellations of the complex plane

2008-11-11

arxiv.org/abs/0811.1678v1

Topology and its Applications 157 (2010), 1873-1899

Mathematics

Geometric Topology

Version 1. 43 pages. 13 .eps figures

Scientific paper

10.1016/j.topol.2010.02.010

To each once-punctured-torus bundle, $T_\phi$, over the circle with pseudo-Anosov monodromy $\phi$, there are associated two tessellations of the complex plane: one, $\Delta(\phi)$, is (the projection from $\infty$ of) the triangulation of a horosphere at $\infty$ induced by the canonical decomposition into ideal tetrahedra, and the other, $CW(\phi)$, is a fractal tessellation given by the Cannon-Thurston map of the fiber group switching back and forth between gray and white each time it passes through $\infty$. In this paper, we study the relation between $\Delta(\phi)$ and $CW(\phi)$.

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