Branched Cauchy-Riemann Structures on Once-Punctured Torus Bundles

2019/02/10 by Casella, Alex
#32V05 #57M50 #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1902.03662

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 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 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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