Mostow rigidity of rank 1 discrete groups with ergodic Bowen–Margulis measure
Mostow rigidity of rank 1 discrete groups with ergodic Bowen–Margulis measure
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DOI：
10.1007/s002220050069
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发表时间：
1996-05
影响因子：
3.1
通讯作者：
C. Yue
中科院分区：
文献类型：
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作者：
C. Yue
本文是我们以前工作[Y1]、[Y2]的另一个续篇。为了正确地理解本文的结果，我们回顾了以前的一些结果。自从Mostow强刚性定理出现以来，人们发展了两种方法来研究半单李群的离散子群的全局刚性。Marguis的遍历理论方法在他的超刚性和算术性定理[MA1]中达到顶峰。由Siu，Yau首先提出并由他人进一步发展的调和映射几何方法(见[Si]，[JY1]，[Sa]，[M]，[c1])也在几何超刚性中达到高潮(Mok-Siu-Yeung[MSY]，另见Jost-Yau[JY2]和Gromov-Schoen[GS])。最近，Besson-Courtois-Gallot[BCG]又出现了一个漂亮的初等处理，它给出了排名第一的莫斯托刚性的一个全新的证明。Marguis的超刚性适用于实数秩为2的非紧半单李群中的格，几何超刚性适用于满足Kazhdan性质的格(包括实数为2的非紧半单李群中的格和SP(n；1)或OO(2；1)中的格)。算术性定理特别暗示了这种李群的绝大多数离散子群是非格子群。虽然我们对晶格所享有的丰富的刚性现象感到惊讶，但我们也惊讶于我们对无限体积的绝大多数离散子群知之甚少。这并不是因为它们不那么重要或不那么美丽(只要想象一下克莱恩群的丰富理论)，而是因为我们缺乏理解，或者更准确地说，我们缺乏探索它们的工具。例如，调和映射法目前显然不适用于无限体积群。
This paper represents another sequel to our previous work [Y1],[Y2]. To put the results of the present paper in perspective, we would like to recall some earlier results.Since the appearance of the Mostow strong rigidity theorem, there developed two approaches to the global rigidity of discrete subgroups of semisimple Lie groups. The ergodic theoretic approach by Margulis culminated in his superrigidity and arithmeticity theorem [Ma1]. The geometric approach via harmonic mapping first started by Siu, Yau and further developed by others (see for example [Si],[JY1],[Sa],[M],[C1]), also reached its climax in the geometric superrigidity (Mok–Siu–Yeung [MSY], see also Jost–Yau [JY2] and Gromov–Schoen [GS]). Recently, there appeared another beautiful elementary treatment by Besson–Courtois–Gallot [BCG] which gives, among other things, a completely new proof of Mostow rigidity in rank one. Margulis’ superrigidity applies to lattices in noncompact semisimple Lie groups of real rank= 2. The geometric superrigidity applies to lattices satisfying the Kazhdan property (these include lattices in noncompact semisimple Lie groups of real rank= 2 and lattices in SP (n; 1) or OO (2; 1)). The arithmeticity theorem implies in particular that the great majority of discrete subgroups of such Lie groups are non-lattices. While we feel amazed by the abundance of rigidity phenomenons enjoyed by the lattices, we are also surprised by how little we know about the vast majority of discrete subgroups of infinite covolume. This is not because they are less important or less beautiful (just imagine the rich theory of Kleinian groups), but because of our lack of understanding, or more precisely our lack of tools to explore them. For example, the harmonic map approach apparently does not work at the moment for groups of infinite covolume