Geometric compactification of moduli spaces of half-translation structures on surfaces

TL;DR

This paper constructs a new geometric compactification of the space of half-translation structures on surfaces, using CAT(0) spaces and asymptotic cone techniques, and compares it to existing current-based compactifications.

Contribution

It introduces the space PMix(S) of mixed structures as a new compactification of half-translation structures, providing a geometric perspective on degenerations.

Findings

PMix(S) is a compactification of PFlat(S).

The compactification uses CAT(0) tree-graded spaces.

Comparison with Duchin-Leininger-Rafi's current-based compactification.

Abstract

In this paper, we give an equivariant compactification of the space PFlat(S) of homothety classes of half-translation structures on a compact, connected, orientable surface S. We introduce the space PMix(S) of homothety classes of mixed structures on S, that are CAT(0) tree-graded spaces in the sense of Drutu and Sapir, with pieces which are R-trees and completions of surfaces endowed with half-translation structures. Endowing Mix(S) with the equivariant Gromov topology, and using asymptotic cone techniques, we prove that PMix(S) is an equivariant compactification of PFlat(S), thus allowing us to understand in a geometric way the degenerations of half-translation structures on S. We finally compare our compactification to the one of Duchin-Leininger-Rafi, based on geodesic currents on S, by the mean of the translation distances of the elements of the covering group of S.