Top dimensional quasiflats in $CAT(0)$ cube complexes
Top dimensional quasiflats in $CAT(0)$ cube complexes
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$CAT(0)$立方体复合体中的顶维拟平面
DOI：
10.2140/gt.2017.21.2281
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发表时间：
2014
期刊：
影响因子：
--
通讯作者：
Jingyin Huang
中科院分区：
文献类型：
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作者：
Jingyin Huang
我们证明了$n$维$cat(0)$立方体复形中的每个$n$-拟平坦与$n$维正交体的有限并有有限的Hausdorff距离。然后，我们引入了一类立方体复形，称为{EM弱特殊}立方体复形，并证明了它们的泛覆盖之间的拟等距保持顶维平坦。利用它，我们建立了直角Artin群的几个拟等距不变量。
我们的一些结果也推广到有限几何维度的$cat(0)$空间。特别地，我们用不同的方法证明了欧氏建筑中的一个顶维拟平坦是Hausdorff逼近于Weyl锥的有限并的事实。
We show that every $n$-quasiflat in a $n$-dimensional $CAT(0)$ cube complex is at finite Hausdorff distance from a finite union of $n$-dimensional orthants. Then we introduce a class of cube complexes, called {\em weakly special} cube complexes and show that quasi-isometries between their universal coverings preserve top dimensional flats. We use this to establish several quasi-isometry invariants for right-angled Artin groups.
Some of our arguments also extend to $CAT(0)$ spaces of finite geometric dimension. In particular, we give a short proof of the fact that a top dimensional quasiflat in a Euclidean buildings is Hausdorff close to finite union of Weyl cones, which was previously established in several other authors by different methods.