The pressure metric for Anosov representations

TL;DR

This paper introduces a new pressure metric for Anosov representations using thermodynamic formalism, establishing its analyticity, rigidity, and invariance properties, and applies it to deformation spaces of hyperbolic groups and Lie groups.

Contribution

It develops a novel intersection notion and pressure metric for Anosov representations, extending geometric structures on deformation spaces with invariance and analyticity results.

Findings

Constructed a $Out( ext{ extGamma})$-invariant Riemannian metric on deformation spaces.

Produced mapping class group invariant metrics on Hitchin components.

Showed Hausdorff dimension varies analytically in families of convex cocompact representations.

Abstract

Using the thermodynamics formalism, we introduce a notion of intersection for projective Anosov representations, show analyticity results for the intersection and the entropy, and rigidity results for the intersection. We use the renormalized intersection to produce a $Out(\Gamma)$-invariant Riemannian metric on the smooth points of the deformation space of irreducible, projective Anosov representations of a word hyperbolic group $\Gamma$ into $SL(m,R)$ whose Zariski closure contains a generic element. In particular, we produce mapping class group invariant Riemannian metrics on Hitchin components which restrict to the Weil--Petersson metric on the Fuchsian loci. Moreover, we produce $Out(\Gamma)$-invariant metrics on deformation spaces of convex cocompact representations into $PSL(2,C)$ and show that the Hausdorff dimension of the limit set varies analytically over analytic families of convex cocompact representations into any rank 1 semi-simple Lie group.