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

摘要

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