Poisson bracket of trace functions on the Fuchsian locus and Wolpert's formulas

Abstract

We develop a systematic method for computing traces of products of M\"obius transformations associated with oriented geodesics on a hyperbolic surface. The method is based on a normalization of matrices in ${SL}_2(\mathbb R)$ which expresses trace identities in terms of hyperbolic lengths, intersection angles, and signed distances along geodesics. Using these trace computations together with Goldman's description of the Atiyah-Bott-Goldman symplectic form on the character variety, we derive explicit geometric formulas for the Poisson brackets of trace functions associated with closed geodesics. More precisely, we express the Poisson bracket of two trace functions and the iterated Poisson bracket of three trace functions in terms of the hyperbolic lengths of the corresponding geodesics, their intersection angles, and the signed distances between intersection points. The results naturally lead to a unified perspective for revisiting Wolpert's cosine and sine formulas and deriving new proofs of them.