Subgroups of Lacunary Hyperbolic Groups and Free Products

TL;DR

This paper characterizes when lacunary hyperbolic groups are stable under free products, studies their subgroup limits, constructs new examples including divisible groups, and explores extensions with property (T) von Neumann algebras.

Contribution

It provides a dynamical criterion for stability under free products and constructs new lacunary hyperbolic groups with specific subgroup and extension properties.

Findings

Characterization of stability under free products

Construction of lacunary hyperbolic groups with prescribed subgroup collections

Extension of groups with property (T) von Neumann algebras

Abstract

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 $(T)$.