Apparaît dans la collection : 2025 - T2 - WS1 - Higher rank geometric structures, Higgs bundles and physics

The space of Hitchin representations of the fundamental group of a closed surface into $\mathrm{SL}(n,\mathbb{R})$ embeds naturally in the space of projective oriented geodesic currents. A classical result in Teichmüller theory is that for $n=2$, currents in the boundary are measured laminations, which are naturally dual to $\mathbb{R}$-trees. In general, we show that currents in the boundary of Hitchin components have combinatorial restrictions on self-intersection which depend on $n$. We introduce a notion of dual space to an oriented geodesic current for which the dual space of a discrete boundary current of the $\mathrm{SL}(n,\mathbb{R})$ Hitchin component is a polyhedral complex of dimension at most $n-1$.