The volumes of the Hitchin-Riemann moduli spaces are infinite

Abstract

In this study, we prove that the actions of the mapping class groups on a large range of higher Teichm\"uller spaces with a rank of at least two possess infinite Atiyah-Bott-Goldman covolume. This result encompasses $\mathsf{G}$-Hitchin components of a higher rank split real form $\mathsf{G}$ and each component of the space of $\mathsf{Sp}_{2n}(\mathbb{R})$-maximal representations where $n \geq 2$. To achieve this outcome, we employ Goldman flows to find an infinite series of subsets of identical volume, the images of which in the quotient space are all mutually disjoint.