Cusped Borel Anosov representations with positivity

TL;DR

This paper characterizes certain cusped Borel Anosov representations, showing they are Hitchin if they contain a specific unipotent element, and explores their structural limitations when amalgamated.

Contribution

It establishes a criterion for cusped Borel Anosov representations to be Hitchin based on unipotent elements and analyzes their behavior under amalgamation.

Findings

Cusped Borel Anosov representations with a unipotent element are Hitchin.

Amalgamation of Hitchin and non-Hitchin cusped Borel Anosov representations is not cusped Borel Anosov.

Characterization of the structure of cusped Borel Anosov representations.

Abstract

We show that if a cusped Borel Anosov representation from a lattice $\Gamma \subset \mathsf{PGL}_2(\mathbb{R})$ to $\mathsf{PGL}_d(\mathbb{R})$ contains a unipotent element with a single Jordan block in its image, then it is necessarily a (cusped) Hitchin representation. We also show that the amalgamation of a Hitchin representation with a cusped Borel Anosov representation that is not Hitchin is never cusped Borel Anosov.