This paper proves a clear geometric surprise: when n>2, the thick part of the PSL_n(R)-Hitchin–Riemann moduli space has infinite total volume for any fixed thickness parameter ε>0. In plain terms, a natural region of this higher-rank moduli space — the set of representations that keep the lengths of all essential closed curves uniformly bounded below by ε — cannot have a finite total measure for the standard Atiyah–Bott–Goldman volume form.

The objects involved are generalizations of familiar ones from surfaces and Teichmüller theory. The Hitchin component Hit_n(S) is a higher-rank analogue of Teichmüller space: it parametrizes certain PSL_n(R) representations of the surface group of a closed orientable surface S of genus g>1. The mapping class group acts on this component, and the quotient is the Hitchin–Riemann moduli space M_n(S). For a representation ρ, the paper uses a length function ℓ_ρ defined by the largest and smallest eigenvalues of ρ(γ): ℓ_ρ(γ)=log(ω1(ρ(γ))/ωn(ρ(γ))). The ε-thick part means the set of representations where every essential curve has length >ε.

To show infinite volume the authors build explicit, disjoint families of subsets inside this ε-thick part that each have the same positive volume. Their construction combines two tools. First, a global parametrization of the Hitchin component (from prior work of Bonahon–Dreyer and Zhang) lets them control internal parameters that describe a representation. Second, they use Goldman flows — deformations of a representation obtained by cutting along a curve and regluing with a twist — in particular a ‘‘bulging’’ one-parameter flow that preserves the Atiyah–Bott–Goldman volume. They also use Zhang’s notion of internal sequences: sequences of representations whose certain internal parameters escape every compact set while keeping other spectral gaps uniformly bounded. A key fact they use is that for an internal sequence the quantity K(ρ) (measuring minimal length contribution from crossings) tends to infinity, so many curve lengths go to infinity while specified lengths remain bounded. Combining these ideas they produce infinitely many disjoint images in the moduli space, each with the same volume, forcing the total volume to diverge.