Strong hyperbolicity
Strong hyperbolicity
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DOI：
10.4171/ggd/372
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发表时间：
2014-08
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通讯作者：
B. Nica;Ján Špakula
中科院分区：
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作者：
B. Nica;Ján Špakula
我们提出了超曲性的度量概念，作为获得具有尖锐附加性质的双曲性的一种方法。具体地说，超曲空间是格罗莫夫双曲空间，它们在无穷远处的度量性质与CAT(−1)空间一样好，并且在弱测地线假设下，它们也是强曲的。我们证明了CAT(−1)空间是双曲的。在此过程中，我们确定了H的最佳双曲性常数。我们还证明了由双曲群上的随机游动定义的格林度量是双曲的。在边界处的一个测度论推论是，由随机游动定义的调和测度是视觉Hausdorff测度。
We propose the metric notion of superbolicity as a way of obtaining hyperbolicity with sharp additional properties. Specifically, superbolic spaces are Gromov hyperbolic spaces that are as metrically well-behaved at infinity as CAT(−1) spaces, and, under weak geodesic assumptions, they are strongly bolic as well. We show that CAT(−1) spaces are superbolic. On the way, we determine the best constant of hyperbolicity for H. We also show that the Green metric defined by a random walk on a hyperbolic group is superbolic. A measuretheoretic consequence at the boundary is that the harmonic measure defined by a random walk is a visual Hausdorff measure.