论文标题
脱骨双曲线组和免费产品的亚组
Subgroups of Lacunary Hyperbolic Groups and Free Products
论文作者
论文摘要
如果其渐近锥之一是$ \ mathbb {r} $ -Tree，则有限生成的组是脱纤维。在本文中，我们给出了脱骨双曲线基团的必要条件，以便在自由产品下稳定，通过对脱骨双曲线组的动态表征进行稳定。同样，我们研究了基本亚组的限制，是腔体hperbolic群体的亚组，并将其表征。鉴于任何可计数的基本组的汇总集合，我们表明存在一个玻璃双曲线群，其所有最大亚组的集合是给定的集合。结果，我们构建了一个有限生成的可分裂群体。第一个这样的示例是由V. Guba在\ cite {gu86}中构建的。在第5节中，我们表明，鉴于任何有限生成的组$ q $和一个非基本双曲线$ h $，存在一个简短的精确序列$ 1 \ rightarrow n \ rightarrow g \ rightarrow g \ rightarrow q \ rightarrow q \ rightarrow 1 $，其中$ g $是lacunary超元组，$ n $是$ n $的非基本词。我们的方法允许恢复\ cite [theorem 3] {as14}。在第6节中，我们扩展了$ \ mathcal {r} ip _ {\ Mathcal {t}}}（q）$ \ cite {cdk19}中考虑的$的类别，因此给出了更多的属性$（t）$ von noumann代数的新示例，这些代数具有最大的Voneumann subalgebras $ $（t）。
A finitely generated group is lacunary hyperbolic if one of its asymptotic cones is an $\mathbb{R}$-tree. In this article we give a necessary and sufficient condition on lacunary hyperbolic groups in order to be stable under free product by giving a dynamical characterization of lacunary hyperbolic groups. Also we studied limits of elementary subgroups as subgroups of lacunary hperbolic groups and characterized them. Given any countable collection of increasing union of elementary groups we show that there exists a lacunary hyperbolic group whose set of all maximal subgroups is the given collection. As a consequence we construct a finitely generated divisible group. First such example was constructed by V. Guba in \cite{Gu86}. In section 5 we show that given any finitely generated group $Q$ and a non elementary hyperbolic group $H$, there exists a short exact sequence $1\rightarrow N\rightarrow G\rightarrow Q\rightarrow 1$, where $G$ is a lacunary hyperbolic group and $N$ is a non elementary quotient of $H$. Our method allows to recover \cite[Theorem 3]{AS14}. In section 6, we extend the class of groups $\mathcal{R}ip_{\mathcal{T}}(Q)$ considered in \cite{CDK19} and hence give more new examples of property $(T)$ von Neumann algebras which have maximal von Neumann subalgebras without property $(T)$.
