Gluing equations for $\mathrm{PGL}(n,\mathbb{C})$–representations of $3$–manifolds

Abstract

Garoufalidis, Thurston and Zickert parametrized boundary-unipotent representations of a 3–manifold group into $SL\left(n,ℂ\right)$ using Ptolemy coordinates, which were inspired by $\mathcal{A}$–coordinates on higher Teichmüller space due to Fock and Goncharov. We parametrize representations into $PGL\left(n,ℂ\right)$ using shape coordinates, which are a $3$–dimensional analogue of Fock and Goncharov’s $\mathcal{X}$–coordinates. These coordinates satisfy equations generalizing Thurston’s gluing equations. These equations are of Neumann–Zagier type and satisfy symplectic relations with applications in quantum topology. We also explore a duality between the Ptolemy coordinates and the shape coordinates.