property $(T)$.'
author:
- Krishnendu Khan
title: Subgroups of Lacunary Hyperbolic Groups and Free Products
---

\[section\] \[theorem\][Lemma ]{} \[theorem\][Assumption ]{} \[theorem\][Proposition ]{} \[theorem\][Corollary ]{} \[theorem\][Definition ]{} \[theorem\][Definition ]{} \[theorem\][Notation ]{} \[theorem\][Example ]{}

Introduction
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The interplay between general group theory and combinatiorial investigation of specific groups having some geometric structure has been exploited by Gromov by introducing and developing the notion of hyperbolic groups in his celebrated article [@Gr87]. In this influencing work Gromov proposed wide range research program.

Von Neumann introduced the notion of amenable groups in [@vN29] around 1929 and conjectured that a group $G$ is non amenable if and only if $G$ contains a free subgroup of rank $2$ after showing that no amenable group contains a free subgroup of rank $2$. This conjecture is shown to be false by A. Olshanskii in in [@Ol79]. Later in [@Ol93], A. Olshanskii developed small cancellation theory over hyperbolic groups and showed that one can construct non abelian groups without free subgroups by taking quotient of non elementary hyperbolic groups, which can be realized as a direct limit of hyperbolic groups. One can infact get non amenable groups in the limit by starting with property (T) hyperbolic group.

Intuitive 