# Coarse density of subsets of moduli space

INFORMAL GEOMETRY AND DYNAMICS

##### Speaker:

Benjamin Dozier *- Stony Brook University*

I will discuss coarse geometric properties of algebraic subvarieties of the moduli space of Riemann surfaces. In joint work with Jenya Sapir, we prove that such a subvariety is coarsely dense, with respect to either the Teichmuller or Thurston metric, iff it has full dimension in the moduli space. This work was motivated by an attempt to understand the geometry of the image of the projection map from a stratum of abelian or quadratic differentials to the moduli space of Riemann surfaces. As a corollary of our theorem, we characterize when this image is coarsely dense. A key part of the proof of the theorem involves comparing analytic plumbing coordinates at the Deligne-Mumford boundary to hyperbolic/extremal lengths of curves on nearby smooth surfaces.

**via Zoom**: **https://harvard.zoom.us/j/972495373**