Cusped Borel Anosov representations with positivity

Abstract

We show that if a cusped Borel Anosov representation from a lattice $$\Gamma \subset \textsf{PGL}_2({{\,\mathrm{\mathbb {R}}\,}})$$ Γ ⊂ PGL 2 ( R ) to $$\textsf{PGL}_d({{\,\mathrm{\mathbb {R}}\,}})$$ PGL d ( 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.