The word and Riemannian metrics on lattices of semisimple groups
The word and Riemannian metrics on lattices of semisimple groups
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半单群格上的词和黎曼度量
DOI：
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发表时间：
2000
期刊：
影响因子：
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通讯作者：
L. Stolovitch
中科院分区：
文献类型：
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作者：
A. Lubotsky;S. Mozes;M. Raghunathan;H. Iwaniec;Wenzhi Luo;P. Sarnak;L. Stolovitch
设G是秩为<$2的半单李群，Γ是不可约格.Γ有两个自然度量：一个度量继承自周围李群上的黎曼度量和一个相对于某个有限生成元集定义的字度量。证明了D. Kazhdan（cf. Gromov [Gr 2]）证明了这些度量是Lipschitz等价的。证明了Γ的循环子群是虚幂幺的当且仅当它关于Γ的生成元是指数增长的.
Let G be a semisimple Lie group of rank ⩾2 and Γ an irreducible lattice. Γ has two natural metrics: a metric inherited from a Riemannian metric on the ambient Lie group and a word metric defined with respect to some finite set of generators. Confirming a conjecture of D. Kazhdan (cf. Gromov [Gr2]) we show that these metrics are Lipschitz equivalent. It is shown that a cyclic subgroup of Γ is virtually unipotent if and only if it has exponential growth with respect to the generators of Γ.