Generic Properties of Hitchin Representations

TL;DR

This paper proves that for generic Hitchin representations of surface groups into certain Lie groups, the Jordan projections avoid any given hyperplane, leading to strong density results for orbifold representations.

Contribution

It establishes a generic property of Hitchin representations related to Jordan projections and demonstrates strong density of orbifold Hitchin representations in split real Lie groups.

Findings

For generic Hitchin representations, Jordan projections do not intersect any hyperplane.

Generic orbifold Hitchin representations are strongly dense.

Split real forms of many simple Lie groups contain strongly dense orbifold fundamental groups.

Abstract

Let $G$ be a split real form of a complex simple adjoint group whose Weyl group contains $-1$, let $\lambda$ be the Jordan projection of $G$, and let $S$ be a closed orientable surface of genus at least 2. For a $G$-Hitchin representation $\rho$, we define the set $J(\rho):=\{\lambda(\rho(x))\,|\,x\in\pi_1(S)\setminus \{1\}\}$. Choose any hyperplane $H$ in the maximal abelian subalgebra of the Lie algebra of $G$. Our main result shows that, for a generic $G$-Hitchin representation $\rho$, we have $J(\rho)\cap H=\emptyset$. As an application, we prove that generic orbifold Hitchin representations are strongly dense. This extends the result of Long, Reid, and Wolff for the Hitchin representations of surface groups. Our theorem also shows that the split real forms of many simple adjoint Lie groups contain strongly dense orbifold fundamental groups, partially generalizing the work of Breuillard, Guralnick, and Larsen.