Primitive stable representations in higher rank semisimple Lie groups

TL;DR

This paper investigates primitive stable representations of free groups into higher rank semisimple Lie groups, focusing on convex projective structures and positive representations, establishing their primitive stability under certain conditions.

Contribution

It verifies $\sigma_{mod}$-regularity for convex projective structures and positive representations, and proves their primitive stability for surfaces with one boundary component.

Findings

Convex projective structures are $\sigma_{mod}$-regular.

Positive representations are $\sigma_{mod}$-regular.

Holonomies are primitive stable for surfaces with one boundary.

Abstract

We study primitive stable representations of free groups into higher rank semisimple Lie groups and their properties. Let $\Sigma$ be a compact, connected, orientable surface (possibly with boundary) of negative Euler characteristic. We first verify the $\sigma_{mod}$-regularity for convex projective structures and positive representations. Then we show that the holonomies of convex projective structures and positive representations on $\Sigma$ are all primitive stable if $\Sigma$ has one boundary component.