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Unlike in hyperbolic geometry, the monodromy ideal triangulation of ahyperbolic once-punctured torus bundle $M_f$ has no natural geometricrealisation 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 everyhyperbolic once-punctured torus bundle admits a branched CR structure, whosebranch locus is the set of edges of $\mathcal{D}_f$. Furthermore, we explicitlycompute the ramification order around each component of the branch locus andanalyse the corresponding holonomy representations.

Branched Cauchy-Riemann Structures on Once-Punctured Torus Bundles