用半定规划法计算Kazhdan常数
Computing Kazhdan constants by semidefinite programming
摘要：
离散群的Kazhdan常数很难计算，实际常数只知道几类群。通过用计算机求解一个半定规划问题，得到了离散群的Kazhdan常数的一个下界。正的下界意味着群具有性质（T）。我们详细研究了$\tilde{A}_2$ -建筑物上的格。对于$\tilde{A}_2$ -群，我们的数值界看起来与已知的实际常数相同。这表明我们的方法是有效的。对于Ronan，Tits等人研究的一类群$G_1, \cdots, G_4$，我们根据实验结果推测Laplacian的谱隙为$(\sqrt 2-1)^2$.对于$\mathrm{SL}(3,\Bbb Z)$和$\mathrm{SL}(4,\Bbb Z)$，我们得到的Kazhdan常数的下界，分别为0.2155和0.3285，这是比任何其他已知的界限。我们还得到了0.1710作为Steinberg群$\mathrm{St}_3(\Bbb Z)$的Kazhdan常数的下界。
英文摘要：
Kazhdan constants of discrete groups are hard to compute and the actual constants are known only for several classes of groups. By solving a semidefinite programming problem by a computer, we obtain a lower bound of the Kazhdan constant of a discrete group. Positive lower bounds imply that the group has property (T). We study lattices on $\tilde{A}_2$-buildings in detail. For $\tilde{A}_2$-groups, our numerical bounds look identical to the known actual constants. That suggests that our approach is effective. For a family of groups, $G_1, \cdots, G_4$, that are studied by Ronan, Tits and others, we conjecture the spectral gap of the Laplacian is $(\sqrt 2-1)^2$ based on our experimental results. For $\mathrm{SL}(3,\Bbb Z)$ and $\mathrm{SL}(4,\Bbb Z)$ we obtain lower bounds of the Kazhdan constants, 0.2155 and 0.3285, respectively, which are better than any other known bounds. We also obtain 0.1710 as a lower bound of the Kazhdan constant of the Steinberg group $\mathrm{St}_3(\Bbb Z)$.
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