Hausdorff dimension of limit sets and its second derivative, some higher rank situations

Let G be a semi-simple real algebraic group, $G_\mathbb R$ its real points and $G_\mathbb C$ its complex points. Let $\Gamma$ be a discrete subgroup of the real points $G_\mathbb R.$ The purpose of the talk is to explain a general procedure relating (in some situations) the second variation of the Hausdorff dimension of the limit set of $\Gamma,$ when deforming $\Gamma$ inside the complex group $G_\mathbb C,$ with natural geometries (known as pressure forms) of the real characters $X(\Gamma,G_\mathbb R).$ 

 

This is joint work with M. Bridgeman, B. Pozzetti and A. Wienhard