Cross ratios, surface groups, \(\text{PSL}(n,\mathbb R)\) and diffeomorphisms of the circle (Q926551)

The author introduces the definition of a cross ratio defined on quadruples of points in the boudary at infinity of the fundamental group of a surface, and he develops a theory that makes a relation between representations of fundamental groups of surfaces and this cross ratio. A connected component of the space of representations into \(\mathrm{PSL}(n,\mathbb{R})\) -- known as the \textit{\(n\)-Hitchin component} -- is identified with a subset of cross ratios on the boundary at infinity of the fundamental group. More precisely, the author proves the following: Theorem: There exists a bijection \(\Phi\) from the set of \(n\)-Hitchin representations of the fundamental group to the set of rank \(n\) cross ratios. This bijection is such that for any nontrivial element \(\gamma\) of the fundamental group, \[ \ell_{\mathbf{B}}(\gamma)=w_{\rho}(\gamma), \] where \(\ell_{\mathbf{B}}(\gamma)\) is the period of \(\gamma\) with respect to \(\mathbf{B}=\phi(\gamma)\), and \(w_{\rho}(\gamma)\) is the width of \(\gamma\) with resepct to \(\rho\). It follows from previous work of the author that if \(\rho\) is a Hitchin representation and \(\gamma\) a nontrivial element of the fundamental group, then \(\rho(\gamma)\) is real split. The width \(w_{\rho}(\gamma)\) of \(\gamma\) with respect to \(\rho\) is then defined as \[ w_{\rho}(\gamma) \log \left(\left| \frac{\lambda_{\mathrm{max}}(\rho(\gamma))}{\lambda_{\mathrm{min}}(\rho(\gamma))}\right|\right), \] where \(\lambda_{\mathrm{max}}(\rho(\gamma))\) and \(\lambda_{\mathrm{min}}(\rho(\gamma))\) are the eignevalues of respectively the maximum and minimum absolute values of \(\rho(\gamma)\). The author then studies some representations into \(C^{1,h}(\mathbb{T})\rtimes \mathrm{Diff}^k(\mathbb{T})\) associated to cross ratios and he exhibits a ``character variety'' of these representations. The author shows that this character variety contains all \(n\)-Hitchin components as well as the set of negatively curved metrics on the surface.