Asian Journal of Mathematics

Volume 26 (2022)

Number 6

Branched Cauchy–Riemann structures on once-punctured torus bundles

Pages: 777 – 808



Alex Casella (Department of Mathematics, Florida State University, Tallahassee, Fl., U.S.A.)


Unlike in hyperbolic geometry, the monodromy ideal triangulation of a hyperbolic once-punctured torus bundle $M_f$ has no natural geometric realization 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 contained in the union of all 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.


geometric structures, Cauchy, Riemann, torus bundles, ideal triangulations, branching

2010 Mathematics Subject Classification

32V05, 57M50

Received 11 May 2021

Accepted 1 September 2022

Published 27 April 2023