CAT(-1)-spaces, divergence groups and their commensurators
CAT(-1)-spaces, divergence groups and their commensurators
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DOI：
10.1090/s0894-0347-96-00196-8
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发表时间：
1996
影响因子：
3.9
通讯作者：
M. Burger;S. Mozes
中科院分区：
文献类型：
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作者：
M. Burger;S. Mozes
CAT(−1)-空间是一个度量测地空间，其中每个测地三角形都比它在双曲平面([B]，[BriHa]，[Gr])上的比较三角形薄。CAT(−1)-性质是负曲率概念在奇异空间的许多可能的推广之一。CAT(−1)-空间的重要例子包括截面曲率k≤−1的黎曼流形及其凸子集([B-G-S])、度量树和分段双曲胞复([Mou]，[Da]，[Hag]，[Be 1]，[Be 2]，[B-BR])。本文在如下两种情形下证明了群Λ在CAT(−1)-空间上的等距作用的某些超刚性结果：a.群Λ是具有Γ<；Λ<；ComgΓ的局部紧群G的子群，其中Γ<；G是足够大的离散子群，ComgΓ={g∈G：G−1Γg和Γ共享有限指数的子群}是Γ在G中的公约子。群Λ是G：=∏nα=1Gα(kα)中的不可约格，其中每个Gα是定义在局部域kα上的半单代数群。本文所讨论的问题一方面是由G.A.Marguis([Ma])关于Λ的线性表示理论的早期工作，其中A，G是半单群，Γ<；G是格，另一方面是Lubotzky，Moze和Zimmer([L-M-Z])关于树上Λ的等距作用的结果，其中Γ<；Λ<；ComgΓ，G是正则树的自同构群，Γ<；G是格。我们建立超刚性结果的方法是基于Marguis发展的遍历理论方法([Ma]，[Zi 3]，[A‘C-B])。在这一背景下，局部紧群Γ的边界的下列概念将是有用的：设B是标准的Γ空间，其中σ通过保持μ-有限测度类的Borel自同构作用。
A CAT(−1)-space is a metric geodesic space in which every geodesic triangle is thinner than its associated comparison triangle in the hyperbolic plane ([B], [BriHa], [Gr]). The CAT(−1)-property is one among many possible generalizations to singular spaces of the notion of negative curvature. Important examples of CAT(−1)-spaces include Riemannian manifolds of sectional curvature k ≤ −1 and their convex subsets ([B-G-S]), metric trees and piecewise hyperbolic cell complexes ([Mou],[Da],[Hag],[Be 1],[Be 2],[B-Br]). In this paper we establish certain superrigidity results for isometric actions of a group Λ on a CAT(−1)-space in the following two settings: A. The group Λ is a subgroup of a locally compact group G with Γ < Λ < ComGΓ, where Γ < G is a sufficiently large discrete subgroup and ComGΓ = {g ∈ G : g−1Γg and Γ share a subgroup of finite index} is the commensurator of Γ in G. B. The group Λ is an irreducible lattice in G := ∏n α=1Gα(kα), where each Gα is a semisimple algebraic group defined over a local field kα. The issues addressed in this paper are motivated on one hand by earlier work of G.A. Margulis ([Ma]) dealing with the linear representation theory of Λ, where in case A, G is a semisimple group and Γ < G a lattice, and on the other hand by the results of Lubotzky, Mozes and Zimmer ([L-M-Z]) concerning isometric actions of Λ on trees, where Γ < Λ < ComGΓ, G is the group of automorphisms of a regular tree and Γ < G is a lattice. Our approach to establishing superrigidity results is based on ergodic theoretic methods developed by Margulis ([Ma],[Zi 3],[A’C-B]). In this context, the following notion of boundary of a locally compact group Γ will be useful: let B be a standard Borel space on which Γ acts by Borel automorphisms preserving a σ-finite measure class μ.