Cross ratios and identities for higher Teichmüller-Thurston theory

By F. Labourie, Greg McShane. 2006

65 citations recorded by the source corpus.

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

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/5fb8d146d4278d59a695bdf3b814da4b49962dc6

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

Citations in the corpus

AN INVITATION TO HIGHER TEICHMÜLLER THEORY (2019, background): “For any reductive Lie group, the representation variety of a closed surfaceHom( 1(S); G)/G is a symplectic manifold W. M. Goldman [1984]. On Teichmüller space this symplectic structure interacts nicely with Fenchel-Nielsen coordinates.”
Positivity and higher Teichm\"uller theory (2018, methodology): “We refer the reader to [39, 40, 41, 27, 29, 9, 45] for more geometric and dynamical properties of Hitchin representations.”
An invitation to higher Teichm\"uller theory (2018, background): “See also [81, 113, 41] for generalizations of cross-ratio identities in the context of higher Teichmüller spaces.”
Higher Teichmüller Spaces: from SL(2,R) to other Lie groups (2010, background): “In [72] Labourie and McShane establish generalized McShane identities for the crossratios associated to Hitchin representations into PSL(n,R).”
Cross ratios, surface groups, PSL(n,ℝ) and diffeomorphisms of the circle (2007, methodology): “In collaboration with McShane in [ 24 ], we use these relations to generalise McShane’s identities [26].”