多边形化复合体中的立体几何
Cubical Geometry in the Polygonalisation Complex
摘要：
我们介绍了一个曲面的复形，一个立方体复形，其顶点对应于复形。这是一个几何模型的映射类群，它是由Harer，Mosher和Penner的作品的动机。利用翻转图的性质，我们表明，midcubes的复杂性，可以扩展到一个家庭的嵌入式和分离超平面，参数化的弧在表面上。我们研究了这些超平面的交叉图，证明了它是弧复形的拟等距图。我们使用交叉图来证明，一般来说，不同的表面有不同的tanalisation复合物。该复形不是CAT（0），但我们可以将Gromov的链接条件失败的顶点进行复形。这提供了一个工具，证明，一般来说，自同构群的nanalisation复杂的（扩展）映射类群的表面。
英文摘要：
We introduce the polygonalisation complex of a surface, a cube complex whose vertices correspond to polygonalisations. This is a geometric model for the mapping class group and it is motivated by works of Harer, Mosher and Penner. Using properties of the flip graph, we show that the midcubes in the polygonalisation complex can be extended to a family of embedded and separating hyperplanes, parametrised by the arcs in the surface. We study the crossing graph of these hyperplanes and prove that it is quasi-isometric to the arc complex. We use the crossing graph to prove that, generically, different surfaces have different polygonalisation complexes. The polygonalisation complex is not CAT(0), but we can characterise the vertices where Gromov's link condition fails. This gives a tool for proving that, generically, the automorphism group of the polygonalisation complex is the (extended) mapping class group of the surface.
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