The geometry of totally geodesic subvarieties of moduli spaces of Riemann surfaces

TL;DR

This work investigates totally geodesic subvarieties of moduli spaces via Teichmüller theory, proving a semisimplicity property for their boundary in the Deligne–Mumford compactification and showing that boundary components decompose into products of simple factors with diagonal-like metric behavior. At the Teichmüller level, the boundary decomposition yields that these submanifolds and their orbifold fundamental groups are hierarchically hyperbolic, enabling coarse geometric descriptions via subsurface curve graphs. The authors develop an interdisciplinary toolkit spanning dynamics, algebraic geometry, geometric group theory, and both classical and modern Teichmüller theory, creating new rigidity and flexibility phenomena and providing a framework for classifying higher-dimensional totally geodesic subvarieties. They connect invariant subvarieties of quadratic differentials to moduli-space subvarieties, leverage horocycle and geodesic cylinder deformations, and establish HHS structures and group-theoretic consequences for the associated orbifold fundamental groups. The results advance the classification problem for totally geodesic subvarieties and offer powerful tools for studying their coarse geometry, rigidity properties, and subgroup structures within mapping class groups.

Abstract

We prove a semisimplicity result for the boundary, in the corresponding Deligne-Mumford compactification, of a totally geodesic subvariety of a moduli space of Riemann surfaces. At the level of Teichmüller space, this semisimplicity theorem gives that each component of the boundary is a product of simple factors, each of which behaves metrically like a diagonal embedding. Building on this result, we also show that the associated totally geodesic submanifolds of Teichmüller space and orbifold fundamental groups are hierarchically hyperbolic. The proof intertwines in a novel way results and perspectives originating in dynamics, algebraic geometry, geometric group theory, and both classical and modern Teichmüller theory. It establishes both new rigidity and new flexibility for totally geodesic submanifolds and their associated varieties and orbifold fundamental groups and provides a rich set of new tools for the study of these objects.

Figures (6)

• Figure 1: The cover $X\to Y$ in Example \ref{['E:FirstExample']}.
• Figure 2: Horocycle arcs joining pairs of points in Teichmüller space. Arrows represent positive time directions.
• Figure 3: Dotted lines are Teichmüller horocycles and solid straight lines are Teichmüller geodesics.
• Figure 4: If $N$ is the whole Teichmüller space, pinching the curves drawn gives a simple factor which is a point in $\mathcal{T}(U_1)\times \mathcal{T}(U_2)$ for the trivial reason that $\mathcal{T}(U_1)$ and $\mathcal{T}(U_2)$ are points themselves. In this example each curve is only equivalent to itself.
• Figure 5: The path of Lemma \ref{['L:quasigeodesic']}.

Theorems & Definitions (138)

• Theorem 1.1: Wright
• Theorem 1.3
• Example 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 2.1
• Corollary 2.2
• Lemma 2.3
• proof
• proof : Proof of Corollary \ref{['C:Connected']}