## Final Report Summary - HIGHTEICH (Higher Teichmüller-Thurston Theory: Representations of Surface Groups in PSL(n,R).)

The main achievements of the project are in the following related areas

Use of the thermodynamic formalism in the context of Anosov representations. Hitchin representations has been described in the work of the PI as dynamical objects. François Labourie (PI), Andres Samabarino (supported post-doc) with Martin Bridgeman and Dick Canary have studied in depth the thermodynamic properties of Hitchin representations (and more general objects). They have introduced entropy, pressure and intersection in the more general context of Anosov representations

Algebras and representations: François Labourie (PI) has described a universal Poisson algebra, called “swapping algebra" for Hitchin components. This Poisson algebra is describe through an elementary combinatorial description. It reflects the Poisson algebra on the ‘infinite genus surface" and is also universal in the rank of the involved group. This is related to the work of Xin Nie (supported post-doc) on the Quasi-Poisson Goldman formula. Sun Zhe (supported graduate student) has related the swapping algebra to the cluster algebra in the case of SL(3,R).

Margulis space times: Margulis spaces times are quotient of the flat Minkowski 3-dimensional space by the proper action of a free group. François Labourie (PI) with Goldman has described the geometry of the recurrent space like geodesics, showing that this dynamical system is conjugated (up to representation) to recurrent part of the geodesic flow on some non compact hyperbolic surface. Kassel (partially supported researcher) with her collaborators Guéritaud and Danciger have given parametrisation of the moduli space of Margulis space times and elucidated their topology. Sourav Ghosh (supported graduate student) is presently importing the Thermodynamic formalism into this context, expecting to describe a ‘Pressure metric" on the moduli space of these Margulis space times. Some of these results extend to anti-de Sitter a closely related field on which Sara Maloni (supported post-doc) has also obtained some results.

Higgs bundle, minimal surfaces and the analytical theory of character varieties: François Labourie has obtained a uniformisation theorem for rank 2 Hitchin components, (the groups involved are SL(3,R), SP(4,R) and ). The Hitchin components can be parametrized by a point in Teichmüller space and a holomorphic differentiable of a suitable degree. Hence those Hitchin components become Kähler manifolds. Equivalently, the result can be stated as the existence of a unique minimal surface in the corresponding symmetric space. With Richard Wentworth, the PI has obtained result linking the thermodynamic formalism and Higgs bundle in a recent prepublication. In a connected subject, Brice Lousteau (supported post-doc) has described different family of hyperkähler structures on the moduli space of complex projective structures.

Use of the thermodynamic formalism in the context of Anosov representations. Hitchin representations has been described in the work of the PI as dynamical objects. François Labourie (PI), Andres Samabarino (supported post-doc) with Martin Bridgeman and Dick Canary have studied in depth the thermodynamic properties of Hitchin representations (and more general objects). They have introduced entropy, pressure and intersection in the more general context of Anosov representations

Algebras and representations: François Labourie (PI) has described a universal Poisson algebra, called “swapping algebra" for Hitchin components. This Poisson algebra is describe through an elementary combinatorial description. It reflects the Poisson algebra on the ‘infinite genus surface" and is also universal in the rank of the involved group. This is related to the work of Xin Nie (supported post-doc) on the Quasi-Poisson Goldman formula. Sun Zhe (supported graduate student) has related the swapping algebra to the cluster algebra in the case of SL(3,R).

Margulis space times: Margulis spaces times are quotient of the flat Minkowski 3-dimensional space by the proper action of a free group. François Labourie (PI) with Goldman has described the geometry of the recurrent space like geodesics, showing that this dynamical system is conjugated (up to representation) to recurrent part of the geodesic flow on some non compact hyperbolic surface. Kassel (partially supported researcher) with her collaborators Guéritaud and Danciger have given parametrisation of the moduli space of Margulis space times and elucidated their topology. Sourav Ghosh (supported graduate student) is presently importing the Thermodynamic formalism into this context, expecting to describe a ‘Pressure metric" on the moduli space of these Margulis space times. Some of these results extend to anti-de Sitter a closely related field on which Sara Maloni (supported post-doc) has also obtained some results.

Higgs bundle, minimal surfaces and the analytical theory of character varieties: François Labourie has obtained a uniformisation theorem for rank 2 Hitchin components, (the groups involved are SL(3,R), SP(4,R) and ). The Hitchin components can be parametrized by a point in Teichmüller space and a holomorphic differentiable of a suitable degree. Hence those Hitchin components become Kähler manifolds. Equivalently, the result can be stated as the existence of a unique minimal surface in the corresponding symmetric space. With Richard Wentworth, the PI has obtained result linking the thermodynamic formalism and Higgs bundle in a recent prepublication. In a connected subject, Brice Lousteau (supported post-doc) has described different family of hyperkähler structures on the moduli space of complex projective structures.