The Ptolemy field of $3$-manifold-representations

Abstract

The Ptolemy coordinates for boundary-unipotent SL(n,C)-representations of a 3-manifold group were introduced in Garoufalidis-Thurston-Zickert inspired by the A-coordinates on higher Teichmüller space due to Fock and Goncharov. In this paper, 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.

21 pages, 23 figures

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