HIERARCHICALLY HYPERBOLIC SPACES II: COMBINATION THEOREMS AND THE DISTANCE FORMULA
HIERARCHICALLY HYPERBOLIC SPACES II: COMBINATION THEOREMS AND THE DISTANCE FORMULA
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DOI：
10.2140/pjm.2019.299.257
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发表时间：
2019-04-01
影响因子：
0.6
通讯作者：
Sisto, Alessandro
中科院分区：
文献类型：
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作者：
Behrstock, Jason;Hagen, Mark;Sisto, Alessandro
我们引入了一些用于寻找和研究分层双曲空间（HHS）的工具，这是一类丰富的空间，包括曲面的映射类群、具有泰希米勒度量或外尔 - 彼得森度量的泰希米勒空间、直角阿廷群以及任何紧致特殊立方体复形的万有覆盖。我们首先引入一组简化的公理来定义一个HHS。我们证明所有的HHS都满足一个马斯尔 - 明斯基式的距离公式，从而在不依赖马斯尔 - 明斯基分层机制的情况下，得到了映射类群中距离公式的一个新证明。然后我们研究HHS的例子；例如，我们证明当M是一个闭的不可约3 - 流形时，π₁(M)是一个HHS当且仅当它既不是幂零（Nil）也不是可解（Sol）。我们通过证明一个关于HHS的树（以及HH群的图）的一般组合定理来确立这一点。我们还引入了“分层拟凸性”的概念，它在HHS的研究中类似于拟凸性在格罗莫夫双曲空间研究中所起的作用。
We introduce a number of tools for finding and studying hierarchically hyperbolic spaces (HHS), a rich class of spaces including mapping class groups of surfaces, Teichmuller space with either the Teichmuller or Weil-Petersson metrics, right-angled Artin groups, and the universal cover of any compact special cube complex. We begin by introducing a streamlined set of axioms defining an HHS. We prove that all HHS satisfy a Masur-Minsky-style distance formula, thereby obtaining a new proof of the distance formula in the mapping class group without relying on the Masur-Minsky hierarchy machinery. We then study examples of HHS; for instance, we prove that when M is a closed irreducible 3-manifold then pi(1) M is an HHS if and only if it is neither Nil nor Sol. We establish this by proving a general combination theorem for trees of HHS (and graphs of HH groups). We also introduce a notion of "hierarchical quasiconvexity", which in the study of HHS is analogous to the role played by quasiconvexity in the study of Gromov-hyperbolic spaces.