Computing Kazhdan Constants by Semidefinite Programming
Computing Kazhdan Constants by Semidefinite Programming
复制标题
通过半定规划计算 Kazhdan 常数
DOI：
10.1080/10586458.2017.1396509
复制
发表时间：
2018
影响因子：
0.5
通讯作者：
Kabaya Yuichi
中科院分区：
文献类型：
--
作者：
Fujiwara Koji;Kabaya Yuichi
离散群的Kazhdan常数很难计算，实际的常数只对几类群是已知的。通过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。正下界意味着群具有性质(T)。我们详细地研究了A˜2-建筑上的格。对于A˜2-群，我们的数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。对于组的家族，G1，…，G4，根据我们的实验结果推测拉普拉斯函数的光谱带隙是基于我们的实验结果。并且，我们得到了Kazhdan常数的下界，分别为0.2155和0.3285，它比其他任何已知的界都要好。我们还得到了0.1710作为Steinberg群的Kazhdan常数的下界。
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 A˜2-buildings in detail. For 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,G1, …,G4, that are studied by Ronan, Tits, and others, we conjecture the spectral gap of the Laplacian isbased on our experimental results. Forand, 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.