On the growth of quotients of Kleinian groups
On the growth of quotients of Kleinian groups
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DOI：
10.1017/s0143385710000131
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发表时间：
2010-05
影响因子：
0.9
通讯作者：
F. Dal'bo;M. Peigné;J. Picaud;Andrea Sambusetti
中科院分区：
文献类型：
--
作者：
F. Dal'bo;M. Peigné;J. Picaud;Andrea Sambusetti
本文研究了Kleinian群G（即具有收缩负曲率的Cartan-Hadamard流形的等距离散无挠群）的扩张性和发散性。也就是说，我们给出了保证G的商群$\bar {G}$发散的一般判据和“临界间隙性质”$\delta _{\bar {G}}\lt \delta _G$。作为推论，我们证明了每个满足抛物间隙条件（即对G的每个抛物子群P，δP<δG）的几何有限Kleinian群是增长紧的.这些商群自然地作用在Cartan-Hadamard流形的非单连通分支上，所以Patterson-Sullivan理论的经典论证在这里是不可用的;这迫使我们采用一种更初等的方法，作为副产品，在单连通的情况下产生一个新的几何有限群发散的经典结果的初等证明。构造了Kleinian群的等价类的一些例子，并讨论了结果的最优性.
Abstract We study the growth and divergence of quotients of Kleinian groups G (i.e. discrete, torsionless groups of isometries of a Cartan–Hadamard manifold with pinched negative curvature). Namely, we give general criteria ensuring the divergence of a quotient group $\bar {G}$ of G and the ‘critical gap property’ $\delta _{\bar {G}}\lt \delta _G$. As a corollary, we prove that every geometrically finite Kleinian group satisfying the parabolic gap condition (i.e. δP<δG for every parabolic subgroup P of G) is growth tight. These quotient groups naturally act on non-simply connected quotients of a Cartan–Hadamard manifold, so the classical arguments of Patterson–Sullivan theory are not available here; this forces us to adopt a more elementary approach, yielding as by-product a new elementary proof of the classical results of divergence for geometrically finite groups in the simply connected case. We construct some examples of quotients of Kleinian groups and discuss the optimality of our results.