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

TL;DR

This paper proves a semisimplicity property for the boundary of totally geodesic subvarieties in moduli spaces of Riemann surfaces, revealing their product structure and hierarchical hyperbolicity, with implications across geometry and dynamics.

Contribution

It establishes a novel semisimplicity theorem for the boundary of these subvarieties and demonstrates their hierarchical hyperbolic structure, integrating diverse mathematical perspectives.

Findings

Boundary components are products of simple factors

Totally geodesic submanifolds are hierarchically hyperbolic

Provides new tools for studying moduli spaces and their subvarieties

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 Teichm\"uller 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 Teichm\"uller 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 Teichm\"uller 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.