发表机构
ETH Zurich; University of Zurich（苏黎世联邦理工学院; 苏黎世大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了Stuck–Zimmer猜想的两个新情形，包括不可约格及含SL2(R)因子的群，通过投影稳定子的局部谱隙建立新的稳定子刚性判据。
AI 中文摘要
我们证明了Stuck–Zimmer猜想的两个新情形。首先，我们解决了连通半单实李群（具有有限中心、无紧因子且实秩至少为二）中不可约格（既包括均匀格也包括非均匀格）的遍历概率保持作用的Stuck–Zimmer猜想。此外，当其中一个单因子局部同构于$\mathrm{SL}_2(\mathbb R)$时，我们将Stuck–Zimmer定理推广到这类群的所有不可约概率保持作用。特别地，这解决了$\mathrm{SL}_2(\mathbb{R}) \times \mathrm{SL}_2(\mathbb{R})$的不可约作用的Stuck–Zimmer猜想。证明依赖于一个新的稳定子刚性判据，该判据利用单个简单因子中投影稳定子的局部谱隙。这里的局部谱隙按Boutonnet、Ioana和Salehi Golsefidy的意义理解。
英文摘要
We prove two new cases of the Stuck--Zimmer conjecture. First, we settle the Stuck--Zimmer conjecture for ergodic probability preserving actions of irreducible lattices (both uniform and non-uniform) in connected semisimple real Lie groups with finite center, no compact factor and real rank at least two. We furthermore extend the Stuck--Zimmer theorem to all irreducible probability preserving actions of such groups whenever one simple factor is locally isomorphic to $\mathrm{SL}_2(\mathbb R)$. In particular, this settles the Stuck--Zimmer conjecture for irreducible actions of $\mathrm{SL}_2(\mathbb{R}) \times \mathrm{SL}_2(\mathbb{R})$. The proof relies on a new stabilizer rigidity criterion using local spectral gap of projected stabilizers in a single simple factor. Here local spectral gap is understood in the sense of Boutonnet, Ioana and Salehi Golsefidy.
Comments15 pages; comments welcome!
