The Ptolemy field of 3–manifold representations

Abstract

The Ptolemy coordinates for boundary-unipotent SL.n; C/-representations of a 3manifold group were introduced by Garoufalidis, Thurston and Zickert [10] inspired by the A-coordinates on higher Teichmller space due to Fock and Goncharov. We define the Ptolemy field of a (generic) PSL.2; C/-representation and prove that it coincides with the trace field of the representation. This gives an efficient algorithm to compute the trace field of a cusped hyperbolic manifold.