论文标题
准树的近似
Tree approximation in quasi-trees
论文作者
论文摘要
在本文中，我们研究了准树的几何特性，并证明了一些等效标准。我们给出了一棵树的一般结构，该树近似于大地测量空间的末端，并用它来证明每个准树是$（1，c）$ - Quasi-iSi-Imometrict to Simplicial树。结果，我们表明，在准树的情况下，可以改善双曲线空间的Gromov树的近似引理，以给出任何与基数无关的任何点的均匀近似值。由此我们表明，有限子集具有统一的树近似相当于能够通过树均匀地近似整个空间。另一个结果，我们注意到准树的边界是在视觉度量的某种选择下与其近似树的边界的等距，并且这给出了树在树边界上的标准度量的自然扩展。
In this paper we investigate the geometric properties of quasi-trees, and prove some equivalent criteria. We give a general construction of a tree that approximates the ends of a geodesic space, and use this to prove that every quasi-tree is $(1,C)$-quasi-isometric to a simplicial tree. As a consequence, we show that Gromov's tree approximation lemma for hyperbolic spaces can be improved in the case of quasi-trees to give a uniform approximation for any set of points, independent of cardinality. From this we show that having uniform tree approximation for finite subsets is equivalent to being able to uniformly approximate the entire space by a tree. As another consequence, we note that the boundary of a quasi-tree is isometric to the boundary of its approximating tree under a certain choice of visual metric, and that this gives a natural extension of the standard metric on the boundary of a tree.
