On uniform exponential growth for linear groups
On uniform exponential growth for linear groups
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关于线性群的均匀指数增长
DOI：
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发表时间：
2001
期刊：
影响因子：
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通讯作者：
H. Oh
中科院分区：
文献类型：
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作者：
A. Eskin;S. Mozes;H. Oh
设Γ是一个n-生成群。给定一个有限的生成元集S，元素γ ∈ Γ的字长lS（γ）定义为存在s1，· · ·，sn ∈ S <$S −1使得γ = s1 · · · sn的最小正整数。对于每个n ∈ N，用BS（n）表示Γ中相对于S的字长至多为n的元素的集合。由lS（·）的次可加性可推出limn→∞|学士学位（n）|1/n存在，记为ωS（Γ）。若ωS（Γ）> 1，则称一个n阶生成群Γ是指数增长的，若对某个c > 0且d ∈ N，则称它是多项式增长的，|学士学位（n）|≤ c ·nd，对任意n ≥ 1，否则，对Γ的某个有限生成元集S，是中间增长的。观察到，Γ的增长类型不依赖于生成集S的选择。如果一个n阶生成群Γ是线性的，则已知Γ是多项式增长的，在这种情况下Γ是幂零的，否则是指数增长的（[Tit 72]，[Mil 68]，[Wol 68]）。
Let Γ be a finitely generated group. Given a finite set of generators S of Γ, the word length lS(γ) for an element γ ∈ Γ is defined to be the smallest positive integer for which there exist s1, · · · , sn ∈ S ∪ S −1 such that γ = s1 · · · sn. For each n ∈ N, denote by BS(n) the set of elements in Γ whose word length with respect to S is at most n. It follows from the subadditive property of lS(·) that limn→∞ |BS(n)| 1/n exists, which we denote by ωS(Γ). A finitely generated group Γ is said to be of exponential growth if ωS(Γ) > 1, of polynomial growth if for some c > 0 and d ∈ N, |BS(n)| ≤ c · n d for all n ≥ 1 and of intermediate growth otherwise, for some finite generating set S of Γ. Observe that the growth type of Γ does not depend on the choice of generating set S. If a finitely generated group Γ is linear, it is known that Γ is either of polynomial growth in which case Γ is virtually nilpotent, or of exponential growth otherwise ([Tit72], [Mil68], [Wol68]).