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

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.