曲面上半平移结构的模空间的几何紧化
Geometric compactification of moduli spaces of half-translation structures on surfaces
摘要：
本文给出了紧致、连通、可定向曲面S上半平移结构的同素类空间PFlat（S）的等变紧化。本文引入了S上混合结构的同素类空间PMix（S），即Drutu和Sapir意义下的CAT（0）树分次空间，其中的部分是R-树和具有半平移结构的曲面的补集.赋予Mix（S）等变Gromov拓扑，利用渐近锥面技巧，证明了PMix（S）是PFlat（S）的等变紧化，从而使我们能够从几何上理解S上半平移结构的退化.最后，我们用S的覆叠群的元素的平移距离的平均值，将我们的紧化与Duchin-Leininger-Rafi基于测地流的紧化进行比较。
英文摘要：
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.
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