Boundary Currents of Hitchin Components

Abstract

The space of Hitchin representations of the fundamental group of a closed surface $S$ into ${\text{SL}}_n\mathbb{R}$ embeds naturally in the space of projective oriented geodesic currents on $S$. We find that currents in the boundary have combinatorial restrictions on self-intersection which depend on $n$. We define a notion of dual space to an oriented geodesic current, and show that the dual space of a discrete boundary current of the ${\text{SL}}_n\mathbb{R}$ Hitchin component is a polyhedral complex of dimension at most $n-1$. For endpoints of cubic differential rays in the ${\text{SL}}_3\mathbb{R}$ Hitchin component, the dual space is the universal cover of $S$, equipped with an asymmetric Finsler metric which records growth rates of trace functions.

60 pages, 20 figures