Anosov groups are a class of discrete subgroups of semi-simple algebraic groups

analogue to what is known as \emph{convex-co-compact groups} in negative curvature.

Thermodynamical constructions equip the (regular points of the) moduli space of

Anosov representations from $\Gamma$ to $G$ with natural positive semi-definite

bi-linear forms, known as pressure forms. Determining whether such a pressure form

is Riemannian requires non-trivial work.

The purpose of the lecture is to explain some geometrical meaning of these forms,

via a higher rank version of a celebrated result for quasi-Fuchsian space by

Bridgeman-Taylor and McMullen on the Hessian of Hausdorff dimension on pure bending

directions. This is work in collaboration with M. Bridgeman, B. Pozzetti and A.

Wienhard.

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