Purely exponential growth of cusp-uniform actions
Purely exponential growth of cusp-uniform actions
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DOI：
10.1017/etds.2017.37
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发表时间：
2016-02
影响因子：
0.9
通讯作者：
Wen-yuan Yang
中科院分区：
文献类型：
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作者：
Wen-yuan Yang
设可数群G在双曲空间（X，d）上有尖点一致作用，使得G是发散型的.本文的主要结果是由Dal'bo，Otal和Peigné [Séries de Poincaré des groupes géométriquement]引入的条件刻画了轨道增长函数的纯指数增长型。Israel J. Math.118（3）（2000），109-124]。对于几何有限的具有收缩负曲率的Cartan-Hadamard流形，这个条件保证了Bowen-Margulis-Sullivan测度的有限性。在这种情况下，我们的结果恢复了Roblin的一个定理（以粗略的形式）。我们的主要工具是在Gromov边界上的Patterson-Sullivan测度，以及Sullivan阴影引理的一个变体，称为部分阴影引理。这使我们能够证明锥、部分锥或horoball的纯指数增长也等价于Dal'bo-Otal-Peigné条件。这些结果在本文[W. Yang，Patterson-Sullivan测度与相对双曲群的增长性预印本，2013，arXiv：1308.6326]。
Suppose that a countable group $G$ admits a cusp-uniform action on a hyperbolic space $(X,d)$ such that $G$ is of divergent type. The main result of the paper is characterizing the purely exponential growth type of the orbit growth function by a condition introduced by Dal’bo, Otal and Peigné [Séries de Poincaré des groupes géométriquement finis. Israel J. Math. 118(3) (2000), 109–124]. For geometrically finite Cartan–Hadamard manifolds with pinched negative curvature, this condition ensures the finiteness of Bowen–Margulis–Sullivan measures. In this case, our result recovers a theorem of Roblin (in a coarse form). Our main tool is the Patterson–Sullivan measures on the Gromov boundary of $X$ , and a variant of the Sullivan shadow lemma called the partial shadow lemma. This allows us to prove that the purely exponential growth of either cones, or partial cones or horoballs is also equivalent to the Dal’bo–Otal–Peigné condition. These results are used further in a paper by the present author [W. Yang, Patterson–Sullivan measures and growth of relatively hyperbolic groups. Preprint, 2013, arXiv:1308.6326].