高秩acylindricity的半简单理论
The semi-simple theory of higher rank acylindricity
AI总结：
本文提出了一种新的非正曲率群概念，揭示了此类群的强规范积分解，并部分解决了Sela的猜想。
摘要：
我们提出了一种新的非正曲率群的概念：离散可数群在有限个δ-超几何空间的有限积上（AU-acylindrically）作用，且具有通用类型因子。受经典（S-算术）晶格理论和蓬勃发展的acylindrically超几何群理论的启发，我们证明，此类群在虚拟同构意义下享有强规范积分解。这种半简单分解也降维到外自同构群，使我们能够部分解决Sela最近的一个猜想。我们还开发了各种结构结果，包括自由与阿贝尔蒂茨替代，以及与晶格包络的关系。在过程中，我们给出了关于acylindricity的各种表示论证明——某些方法即使在秩-1情况下也是新的。这种群类的广阔性通过识别其包含例如S-算术晶格（秩-1因子）、acylindrically超几何群、HHGs、具有性质（QT）的群，并在直接积、通过（完全通用类型）子群和有限指数上群等方面封闭来展示。
英文摘要：
We present a new notion of non-positively curved groups: the collection of discrete countable groups acting (AU-)acylindrically on finite products of $δ$-hyperbolic spaces with general type factors. Inspired by the classical theory of ($S$-arithmetic) lattices and the flourishing theory of acylindrically hyperbolic groups, we show that, up to virtual isomorphism, finitely generated groups in this class enjoy a strongly canonical product decomposition. This semi-simple decomposition also descends to the outer-automorphism group, allowing us to give a partial resolution to a recent conjecture of Sela. We also develop various structure results including a free vs abelian Tits' Alternative, and connections to lattice envelopes. Along the way we give representation-theoretic proofs of various results about acylindricity -- some methods are new even in the rank-1 setting. The vastness of this class of groups is exhibited by recognizing that it contains, for example, $S$-arithmetic lattices with rank-1 factors, acylindrically hyperbolic groups, HHGs, groups with property (QT), and is closed under direct products, passing to (totally general type) subgroups, and finite index over-groups.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
