Cross ratios, surface groups, PSL(n,ℝ) and diffeomorphisms of the circle

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Anosov representations: domains of discontinuity and applications (2011, methodology): “In recent years, due to work of Labourie [57–59] and of Fock and Goncharov [29], it has been shown that representations in Hitchin components have beautiful geometric properties, which generalize properties of classical Teichmüller space.”
Higher Teichmüller Spaces: from SL(2,R) to other Lie groups (2010, background): “In particular, he associates a generalized crossratio function to any Hitchin representation into PSL(n,R) and shows that crossratio functions arising from a Hitchin representation into PSL(n,R) are characterized by explicit functional equations [69].”
AN INVITATION TO HIGHER TEICHMÜLLER THEORY (2019, unknown): “François Labourie (2007a).”