The Thick Part of the $\mathrm{PSL}_n(\mathbb{R})$-Hitchin-Riemann Moduli Space has Infinite Volume

Abstract

We prove that the thick part of the $\mathrm{PSL}_n(\mathbb{R})$-Hitchin-Riemann moduli space has infinite total Atiyah--Bott--Goldman volume for $n>2$. This result stands in contrast to Mumford's compactness criterion. To achieve this result, we employ Goldman flows and internal sequences to find an infinite series of subsets of identical volume, the images of which in the Hitchin-Riemann moduli space are all mutually disjoint and sit in the thick part.