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The volumes of the Hitchin-Riemann moduli spaces are infinite

Suhyoung Choi, Hongtaek Jung

TL;DR

The paper proves that the Mod(S)-action on higher Teichmüller spaces associated with split real groups (Hitchin components) and Sp_{2n}(R) maximal representations has infinite total volume when the real rank is at least two. The authors develop Goldman flows and algebraic bending to produce an infinite family of pairwise disjoint open sets of equal volume inside Hit_G(S) and Max_{2n}(S), which descend to disjoint Mod(S)-orbits in the Hitchin-Maximal moduli spaces. A central technical achievement is the properness of algebraic bending, together with collar lemmas and a careful decomposition along essential curves, enabling control over the length spectra and Dehn-twist actions. The results extend to relative settings with boundary and suggest a broad, unifying pattern for higher Teichmüller spaces via Θ-positivity, while raising questions about thick parts and generalizations to other Θ-positive spaces.

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 $n \geq 2$. 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.

The volumes of the Hitchin-Riemann moduli spaces are infinite

TL;DR

The paper proves that the Mod(S)-action on higher Teichmüller spaces associated with split real groups (Hitchin components) and Sp_{2n}(R) maximal representations has infinite total volume when the real rank is at least two. The authors develop Goldman flows and algebraic bending to produce an infinite family of pairwise disjoint open sets of equal volume inside Hit_G(S) and Max_{2n}(S), which descend to disjoint Mod(S)-orbits in the Hitchin-Maximal moduli spaces. A central technical achievement is the properness of algebraic bending, together with collar lemmas and a careful decomposition along essential curves, enabling control over the length spectra and Dehn-twist actions. The results extend to relative settings with boundary and suggest a broad, unifying pattern for higher Teichmüller spaces via Θ-positivity, while raising questions about thick parts and generalizations to other Θ-positive spaces.

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 -Hitchin components of a higher rank split real form and each component of the space of -maximal representations where . 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.
Paper Structure (25 sections, 27 theorems, 180 equations, 1 figure, 1 table)

This paper contains 25 sections, 27 theorems, 180 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Hitchin and maximal representations
  3. Lie groups
  4. Hitchin components
  5. Maximal representations
  6. Limit maps and positivity
  7. Action of the mapping class group
  8. Symplectic structure
  9. Goldman flows
  10. Decomposition of Hitchin and maximal components
  11. When $\alpha$ is non-separating
  12. When $\alpha$ is separating
  13. Algebraic bending
  14. Normalized representations
  15. Algebraic bending
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $S$ be a closed orientable surface of genus $>1$, and let $\mathsf{G}$ be a split real form of a complex simple adjoint group. The total volume of the $\mathsf{G}$-Hitchin-Riemann moduli space $\mathcal{M}_{\mathsf{G}}(S)$ with respect to the volume form $\mathrm{vol}$ is infinite provided the r

Figures (1)

  • Figure 1: Enumeration of simple roots

Theorems & Definitions (49)

  • Theorem 1.1: Main Theorem
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1: zhang2015
  • Remark 2.2
  • Theorem 2.3: Theorems 11 and 14 of guichard2010
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 39 more