On the spectrum of the sum of generators for a finitely generated group
On the spectrum of the sum of generators for a finitely generated group
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关于有限生成群的生成元之和的谱
DOI：
10.1007/bf02761298
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发表时间：
1993
影响因子：
1
通讯作者：
A. Valette
中科院分区：
文献类型：
--
作者：
Pierre de la Harpe;A. Robertson;A. Valette
设Γ是一个群生成的群。在群代数[r]中，形成r的生成元的有限集合S的平均。给定一个酉表示π，我们将算子π（h）的谱性质与Γ和π的性质联系起来，特别地，对于Γ的泛表示πun，我们证明了如下结果.首先，谱Sp（πun（h））包含模为1的复数z当且仅当Sp（πun（h））乘z不变，当且仅当存在特征
$$\chi：\Gamma \to \mathbb{T}$$
使得η（S）={z}。其次，对于S −1=S，群Γ具有卡日丹性质（T）当且仅当1在Sp（πun（h））中孤立;在这种情况下，1与谱上其他点之间的距离给出了卡日丹常数的下界。许多例子说明了结果。
AbstractLet Γ be a finitely generated group. In the group algebra ℂ[Γ], form the averageh of a finite setS of generators of Γ. Given a unitary representation π of Γ, we relate spectral properties of the operator π(h) to properties of Γ and π.For the universal representationπun of Γ, we prove in particular the following results. First, the spectrum Sp(πun(h)) contains the complex numberz of modulus one iff Sp(πun(h)) is invariant under multiplication byz, iff there exists a character
$$\chi :\Gamma \to \mathbb{T}$$
such that η(S)={z}. Second, forS−1=S, the group Γ has Kazhdan’s property (T) if and only if 1 is isolated in Sp(πun(h)); in this case, the distance between 1 and other points of the spectrum gives a lower bound on the Kazhdan constants. Numerous examples illustrate the results.