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 Teichmüller 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 $n\ge 2$. 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.

37 pages. Extend the main result to all Hitchin components as well as Sp(2n,R)-maximal components