On the Growth of $L^2$-Invariants of Locally Symmetric Spaces, II: Exotic Invariant Random Subgroups in Rank One
On the Growth of $L^2$-Invariants of Locally Symmetric Spaces, II: Exotic Invariant Random Subgroups in Rank One
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关于局部对称空间的 $L^2$-不变量的增长，II：一级中的奇异不变量随机子群
DOI：
10.1093/imrn/rny080
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发表时间：
2018
影响因子：
1
通讯作者：
Samet, Iddo
中科院分区：
文献类型：
--
作者：
Abert, Miklos;Bergeron, Nicolas;Biringer, Ian;Gelander, Tsachik;Nikolov, Nikolay;Raimbault, Jean;Samet, Iddo
在本系列的第1篇论文中，我们研究了李群G中格序列的Betti数、扭挠和其他谱不变量的渐近行为。我们工作的一个关键内容是研究G.任何格序列都有一个子序列收敛到一个IRS，当G的秩较高时，Nevo-Stuck-Zimmer定理将G的所有IRS分类。利用这种分类，可以推导出格的谱不变量的渐近陈述。当G具有真实的秩1时，IRS空间更加复杂。本文在群SO（n，1），n≥ 2中构造了若干不可数的IRS族.当n = 2，3时，我们给出了特定于维度的结构，并且还描述了适用于每个n的一般胶合结构。后一种构造的部分灵感来自Gromov和Piatetski-Shapiro在SO（n，1）中构造非算术格。
In the 1st paper of this series we studied the asymptotic behavior of Betti numbers, twisted torsion, and other spectral invariants for sequences of lattices in Lie groupsG. A key element of our work was the study ofinvariant random subgroups(IRSs) ofG. Any sequence of lattices has a subsequence converging to an IRS, and whenGhas higher rank, the Nevo–Stuck–Zimmer theorem classifies all IRSs ofG. Using the classification, one can deduce asymptotic statements about spectral invariants of lattices. WhenGhas real rank one, the space of IRSs is more complicated. We construct here several uncountable families of IRSs in the groups SO(n, 1),n≥ 2. We give dimension-specific constructions whenn= 2, 3, and also describe a general gluing construction that works for everyn. Part of the latter construction is inspired by Gromov and Piatetski-Shapiro’s construction of non-arithmetic lattices in SO(n, 1).