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

Abstract

Dans cet article, on construit une compactification équivariante de l'espace P Flat(Σ) des classes d'homothétie de structures de demi-translation sur une surface Σ compacte, connexe, orientable. On définit l'espace P Mix(Σ) des classes d'homothétie de structures mixtes sur Σ, qui sont des structures arborescentes, au sens de Drutu et Sapir, CAT(0), dont les pièces sont des arbres réels ou des complétés de surfaces munies de structures de demi-translation.En munissant Mix(Σ) de la topologie de Gromov équivariante, et en utilisant des techniques de cônes asymptotiques à la Gromov, on montre que P Mix(Σ) est une compactification équivariante de P Flat(Σ), ce qui nous permet de comprendre géométriquement les dégénérescences de structures de demi-translation sur Σ. On compare ensuite cette compactification à celle de Duchin-Leininger-Rafi, qui utilise des courants géodésiques, en passant par les distances de translation des éléments du groupe de revêtement de Σ.Abstract : In this paper, we give an equivariant compactification of the space P Flat(Σ) of homothety classes of half-translation structures on a compact, connected, orientable surface Σ. We introduce the space P Mix(Σ) of homothety classes of mixed structures on Σ, 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(Σ) with the equivariant Gromov topology, and using asymptotic cone techniques, we prove that P Mix(Σ) is an equivariant compactification of P Flat(Σ), thus allowing us to understand in a geometric way the degenerations of half-translation structures on Σ. We finally compare our compactification to the one of Duchin-Leininger-Rafi, based on geodesic currents on Σ, by the mean of the translation distances of the elements of the covering group of Σ. 1The goal of this paper is to construct and to describe a geometric compactification, natural under the action of the mapping class group, of the space of homothety classes of halftranslation stuctures on a compact surface, endowed with the equivariant Gromov topology. It is part of the wide field of study of deformations of geometric structures on surfaces (see for instance [Gol]). Let Σ be a compact, connected, orientable surface of genus g > 2, without boundary for simplicity (see [Mor1]) for the general case). After the founding fathers Gauss and Riemann who have studied conformal geometry on surfaces, the Teichmüller spaces T (Σ) of isotopy classes of hyperbolic metrics on Σ, have been studied by for instance Fricke, Klein, Fenchel, Nielsen, and the moduli spaces of real projective structures by for instance Goldman and Choi [CG]. The analysis of the space Flat(Σ) of half-translation structures on a Σ is currently blooming, with the works notably of Calta, Eskin, Hubert, Lanneau, Masur, McMullen, Möller, Myrzakhani, Schmidt, Smillie, Veech, Weiss, Yoccoz and Zorich. When these deformations spaces are non compact, it is important and useful to consider the asymptotic behavior ...

Weyl chamber length compactification of the ${\rm PSL}(2,\mathbb R)\times{\rm PSL}(2,\mathbb R)$ maximal character variety

“…In this section we introduce a natural notion of R 2 -mixed structure on a surface. This generalizes flat structures, and refines the notion of mixed structure introduced by Duchin-Leininger-Rafi and Morzadec [DLR10,Mor18]. The definition follows the point of view of [DLR10], see Section 6 for the metric viewpoint analoguous to [Mor18].…”

“…Notice that in his thesis [Mor18] Morzadec used tree-graded spaces to obtain a geometric compactification of the space of flat structures, and relate them with the mixed structures of [DLR10]. Also, in the case of G = PSL(3, R) and Γ a punctured surface group, the third named author associates in [Par15] to large families of boundary points of Ξ WL Hit (Γ, PSL(3, R)) explicit finite a + -simplicial complexes whose universal cover, a tree-graded space with flat surface pieces, embbeds equivariantly in the building preserving the natural a + -metric.…”

“…Inspired by the work of Duchin-Leiniger-Rafi [DLR10] and of Morzadec [Mor18], we give a geometric interpretation of boundary points in the case of n = 2. Duchin-Leiniger-Rafi introduced mixed structures to give a geometric compactification of the space of Flat(Σ) of half translation structures on a compact surface Σ up to isometry.…”

Holomorphic quadratic differentials in Teichmüller theory

“…For a parallel discussion of translation surfaces corresponding to holomorphic 1forms, and its relation to the theory of billiards, see the surveys [132,133] and [135]. For more about the structure, closed trajectories, length-spectra and degeneration of such singular-flat metrics, see for example, [104], [30], [23], [35] and [97].…”