Filtered screens and augmented Teichm\"uller space

TL;DR

This paper introduces a new bordification of the decorated Teichmüller space using filtered screens, providing a CW decomposition of the augmented space with a novel indexing by partially oriented stratum graphs.

Contribution

It develops a new bordification of decorated Teichmüller space via filtered screens and characterizes conditions for short curves, leading to a CW decomposition of the augmented space.

Findings

Identifies conditions for paths to produce vanishing length curves.

Constructs a quotient space with a CW decomposition.

Introduces partially oriented stratum graphs as an indexing tool.

Abstract

We study a new bordification of the decorated Teichm\"uller space for a multiply punctured surface F by a space of filtered screens on the surface that arises from a natural elaboration of earlier work of McShane-Penner. We identify necessary and sufficient conditions for paths in this space of filtered screens to yield short curves having vanishing length in the underlying surface F. As a result, an appropriate quotient of this space of filtered screens on F yields a decorated augmented Teichm\"uller space which is shown to admit a CW decomposition that naturally projects to the augmented Teichm\"uller space by forgetting decorations and whose strata are indexed by a new object termed partially oriented stratum graphs.