发表机构
University of Luxembourg（卢森堡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究闭曲面射影填充测地流空间上Lipschitz度量的秩与渐近维数，证明其无限，并给出有限单纯形秩的尖锐上界，同时证明Teichmüller空间秩为3g-3，两者非拟等距。
AI 中文摘要
对于亏格至少为2的闭可定向曲面，我们研究了其射影填充测地流空间上（对称化的）Lipschitz度量的大规模几何。我们证明了填充测地流空间的秩以及渐近维数是无限的。这给出了与闭曲面相关的自然固有空间、具有映射类群的度量固有等距作用且具有无限渐近维数的第一个例子。相反，我们证明了任何测地流的有限单纯形的秩一致地有上界$8g-9$。然后我们给出了其秩和渐近维数等于$8g-9$的充分判据，并构造了满足该判据的例子，从而表明该界是尖锐的。我们进一步证明了配备Lipschitz度量的Teichmüller空间的秩等于$3g-3$。特别地，射影填充测地流空间与Teichmüller空间不是拟等距的。
英文摘要
For a closed orientable surface of genus at least 2, we study the large-scale geometry of the (symmetrized) Lipschitz metric on its space of projective filling geodesic currents. We show that the rank and, hence, asymptotic dimension of the space of filling geodesic currents is infinite. This gives the first example of a natural proper space associated to a closed surface, with a metrically proper isometric action of the mapping class group, that has infinite asymptotic dimension. In contrast, we show that any finite simplex of geodesic currents has rank uniformly bounded above by $8g-9$. We then show a sufficient criterion for its rank and asymptotic dimension to be equal to $8g-9$ and construct examples satisfying it, thus showing the bound is sharp. We further prove that Teichmüller space equipped with the Lipschitz metric has rank equal to $3g-3$. In particular, the space of projective filling geodesic currents and the Teichmüller space are not quasi-isometric.
Comments70 pages, 3 figures
