Cross ratios and identities for higher Teichmüller-Thurston theory

Abstract

We generalise the McShane-Mirzakhani identities from hyperbolic geometry to arbitrary cross ratios. We define and study Hitchin representations of open surface groups to $\mathrm{PSL}(n,\mathbb{R})$. We associate cross ratios to these representations and then give explicit expressions for our generalised identities in terms of (a suitable choice of) Fock-Goncharov coordinates