Ends of strata of the moduli space of quadratic differentials

TL;DR

This paper proves that each connected component of the strata in the moduli space of quadratic differentials has only one topological end, using Veech zippered rectangles and Rauzy classes to analyze boundary behavior.

Contribution

It establishes the topological end structure of strata in the moduli space of quadratic differentials, a previously unexplored aspect of their topology.

Findings

Each connected component has only one topological end.

The boundary behavior is analyzed via Veech zippered rectangles.

Uses Rauzy classes to handle complex boundary splitting data.

Abstract

Very few results are known about the topology of the strata of the moduli space of quadratic differentials. In this paper, we prove that any connected component of such strata has only one topological end. A typical flat surface in a neighborhood of the boundary is naturally split by a collection of parallel short saddle connections, but the discrete data associated to this splitting can be quite difficult to describe. In order to bypass these difficulties, we use the Veech zippered rectangles construction and the corresponding (extended) Rauzy classes.