论文标题
关于共轭规范，渐近锥，度量超强和合同
On conjugation invariant norms, asymptotic cones, metric ultraproducts and contractibility
论文作者
论文摘要
在本文中，我们证明了Lemmata关于渐近锥和度量超强的强合同，我们将其应用于有限生成的单词规范和共轭不变规范的情况。我们在免费产品上恢复了经典已知的合同性结果，并证明了配备有结合不变标准的无限对称组$ \ sym_ \ sym_ \ sym_ \ sym_ \ sym_ \ sym_ \ Infty $的合同。此外，我们给出了由一般线性组亚组引起的可缩度度量超强的示例。此外，我们讨论了作为偶然规范的渐近锥的群体的代数特性。例如，我们表明，无限对称组的渐近锥本身是一个代数简单的群体，与Elek和Szabo定义的普遍秘密群体密切相关。
In the present paper we prove lemmata on strong contractibility in asymptotic cones and metric ultraproducts which we apply to both the case of finitely generated word norms and the case of conjugation invariant norms. We recover classically known contractibility results on free products and prove the contractibility of the asymptotic cone of the infinite symmetric group $\Sym_\infty$ equipped with a conjugation invariant norm. Furthermore, we give examples of contractible metric ultraproducts arising from subgroups of general linear groups. Additionally, we discuss algebraic properties of groups arising as asymptotic cones for conjugation invariant norms. For example, we show that the asymptotic cone of the infinite symmetric group is itself an algebraically simple group relating strongly to the universally sophic groups defined by Elek and Szabo.
