有限直积商群与一致Kazhdan常数
Non Uniform Kazhdan Constant for Linear Groups
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中文总结 AI 辅助
该研究证明，存在任意多非平凡有限群直积商群的有限生成群，其一致Kazhdan常数为0，并将此结果应用于SL_n(ℤ)，得出其一致Kazhdan常数为0的结论。
中文摘要 AI 辅助
我们证明，当有限生成群Γ存在任意多非平凡有限群的直积构成的有限商群时，其一致Kazhdan常数κ_u(Γ)会消失。更精确地说，具有m个因子的商群可生成Γ的有限生成集S_m，使得κ(Γ,S_m)≤2/√m。将该准则应用于对不同素数的同时模约化，可得对所有n≥2，κ_u(SL_n(ℤ))=0。
英文摘要
A group $Γ$ generated by a finite set $S$ has Property $(T)$ if the associated Kazhdan constant $κ(Γ,S)$ is positive. If so, the same holds for every finite generating set $S$. There are Property $(T)$ groups which are uniformly $(T)$, and others which are not. It is a well-known problem whether the classical examples of Property $(T)$ groups, $\mathrm{SL}_n(\mathbb{Z})$, $n\geq 3$, are uniformly $(T)$ or not. We show that they are not. Moreover, the same holds for every infinite finitely generated linear group.
