Asymptotic properties of infinitesimal characters and applications

TL;DR

This paper investigates the geometric and algebraic properties of character varieties and Anosov representations, establishing new interiority results and linking pressure forms with symplectic structures, with applications to higher Teichmüller theory.

Contribution

It proves the non-empty interior of the cone of Jordan variations and length-normalized variations, and connects pressure forms to symplectic structures in higher-rank Teichmüller spaces.

Findings

Non-empty interior of the cone of Jordan variations.

Non-empty interior of length-normalized variations for split groups.

Degeneration of Hausdorff dimension governed by a Diophantine equation.

Abstract

Inspired by Benoist, we study objects linked to integrable tangent vectors on the character variety of a semi-group $\Gamma$ with values in a semi-simple real-algebraic group $\mathsf G$. We prove the \emph{cone of Jordan variations} has non-empty interior and, when $\mathsf G$ is split, establish non-empty interior of the set of \emph{length-normalized variations}. We apply these techniques to pressure forms on Anosov representations and higher-rank Teichm\"uller spaces. We identify an explicit functional $\varphi\in\mathfrak a^*$ whose pressure form is compatible with Goldman's symplectic form at Fuchsian points in the Hitchin component. Finally, we show the degeneration of the Hausdorff dimension of \emph{higher}-quasi-circles is governed by a Diophantine equation.