On Mirzakhani's twist torus conjecture

TL;DR

This work proves that expanding families of twist tori in the moduli space $\mathcal{M}_g$ equidistribute to Lebesgue-type measures in a broad range of cases, with the limiting distribution determined by the combinatorics of the gluing and the associated ribbon graphs. The authors leverage a precise earthquake–horocycle conjugacy $\mathcal{O}$ to transfer equidistribution results from the flat setting of $\mathcal{Q}^1\mathcal{M}_g$ to the hyperbolic moduli space, and apply the Eskin–Mirzakhani–Mohammadi framework on affine invariant subvarieties to classify possible limits. For generic choices (where $\nabla_\gamma(\boldsymbol{\ell})=0$), the twist tori push forward to Mirzakhani's invariant measure $\mu_{\mathrm{Mirz}}/b_g$; in cases with $\nabla_\gamma(\boldsymbol{\ell})>0$, the limits are singular to Mirzakhani and arise from Masur–Smillie–Veech measures on quadratic-differential strata. The results extend to broader families of twist tori built from plumbing fixtures and other constructions, illustrating a robust interplay between hyperbolic and flat dynamics in moduli spaces with implications for orbit closures and counting problems in Teichmüller dynamics.

Abstract

We address a conjecture of Mirzakhani about the statistical behavior of certain expanding families of ``twist tori'' in the moduli space of hyperbolic surfaces, showing that they equidistribute to a certain Lebesgue-class measure along almost all sequences. We also identify a number of other expanding families of twist tori whose limiting distributions are mutually singular to Lebesgue.

Figures (7)

• Figure 1: Hyperbolic and flat twist tori. These tori are mapped to each other under $\mathcal{O}$.
• Figure 2: Projections to the length part of Fenchel--Nielsen coordinates. From left to right: the horospheres considered in Mirz_horo, the horospheres considered in spine, the twist tori of Conjecture \ref{['conj:twist tori']}, and a horospherical slice measure on a lower-dimensional stratum.
• Figure 3: A pair of spines that does not appear in $\mathcal{TRG}(S \setminus \gamma)$ and their dual arc systems. Together, the combinatorial conditions imply that the length of $c$ must be 0, which cannot happen inside $\mathcal{T}(S \setminus \gamma)$.
• Figure 4: Left: a pair of multicurves on a surface. Right: An orientable ribbon graph $\Gamma$ with topological type $S\setminus \gamma$ where $(\gamma, \Gamma)$ is not jointly orientable. If one builds a square-tiled surface whose vertical and horizontal cylinders are the pair of multicurves on the left, then its graph of horizontal separatrices is the one shown on the right.
• Figure 5: A regular plumbing fixture on a $7$-holed sphere.

Theorems & Definitions (53)

• Conjecture 1.1: Mirzakhani's twist torus conjecture
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 1.5
• Theorem 2.1
• Remark 2.2
• Theorem 2.3
• Remark 2.4
• Definition 3.1