发表机构
Research Center for Operator Algebras, School of Mathematical Sciences, East China Normal University（华东师范大学数学科学学院算子代数研究中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明对每个$n\geq3$，$\mathrm{SL}_n(\mathbb Z)$的所有两元素生成集的Kazhdan常数下确界为$0$，回答了生成集基数有界时下确界是否仍为零的问题。
AI 中文摘要
对于$n\geq3$，群$\mathrm{SL}_n(\mathbb Z)$具有性质$(T)$，因此每个有限生成集都有正的Kazhdan常数。然而，最近证明这些常数在所有有限生成集上的下确界为$0$。这自然引发一个问题：当生成集的基数预先有界时，下确界是否仍为零。我们证明，对于每个$n\geq3$，$\mathrm{SL}_n(\mathbb Z)$的所有两元素生成集的Kazhdan常数的下确界为$0$。
英文摘要
For $n\geq3$, the group $\mathrm{SL}_n(\mathbb Z)$ has property $(T)$, so every finite generating set has a positive Kazhdan constant. However, it was recently shown that the infimum of these constants over all finite generating sets is $0$. This naturally raises the question of whether the infimum remains zero when the cardinality of the generating sets is bounded in advance. We prove that the infimum of the Kazhdan constants of $\mathrm{SL}_n(\mathbb Z)$ over all two-element generating sets is $0$ for every $n\geq3$.
