Branched Cauchy-Riemann Structures on Once-Punctured Torus Bundles

Alex Casella

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arXiv · 1902.03662

Branched Cauchy-Riemann Structures on Once-Punctured Torus Bundles

Alex Casella

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Abstract

Unlike in hyperbolic geometry, the monodromy ideal triangulation of a hyperbolic once-punctured torus bundle $M_f$ has no natural geometric realisation in Cauchy-Riemann (CR) space. By introducing a new type of $3$--cell, we construct a different cell decomposition $\mathcal{D}_f$ of $M_f$ that is always realisable in CR space. As a consequence, we show that every hyperbolic once-punctured torus bundle admits a branched CR structure, whose branch locus is the set of edges of $\mathcal{D}_f$. Furthermore, we explicitly compute the ramification order around each component of the branch locus and analyse the corresponding holonomy representations.

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Alex Casella. 2019-02-10. Branched Cauchy-Riemann Structures on Once-Punctured Torus Bundles. https://arxiv.org/abs/1902.03662

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