Pressure metrics for cusped Hitchin components

TL;DR

This paper introduces pressure metrics for cusped Hitchin components, extending geometric analysis tools to these representations, including new metrics for the case when d=3 and the Fuchsian group is cocompact.

Contribution

The paper constructs pressure metrics for cusped Hitchin components, including a novel metric for d=3 cases with cocompact Fuchsian groups.

Findings

Pressure metrics are defined for cusped Hitchin components.

A new pressure metric related to the Hilbert length is introduced for d=3.

The metrics generalize previous structures and provide new geometric insights.

Abstract

We study the cusped Hitchin component consisting of (conjugacy classes of) cusped Hitchin representations of a torsion-free geometrically finite Fuchsian group into PSL(d,R). We produce pressure metrics associated to the first fundamental weight and the first simple root. When $d=3$ we produce a pressure metric associated to the Hilbert length, which is new even when the Fuchsian group is cocompact.