On some quotient groups of hyperbolic groups

TL;DR

This paper generalizes Ol'shanskii's results to non-cyclic torsion-free hyperbolic groups, showing they have quotients with all proper subgroups cyclic and non-trivial intersections.

Contribution

It extends Ol'shanskii's work to a broader class of hyperbolic groups, constructing specific quotients with unique subgroup properties.

Findings

Existence of non-Abelian torsion-free quotients with cyclic proper subgroups

Proper subgroups intersect non-trivially

Generalization of Ol'shanskii's results to hyperbolic groups

Abstract

This paper describes some generalizations of the results presented in the book "Geometry of defining Relations in Groups" , of A.Yu.Ol'shanskii to the case of non-cyclic torsion-free hyperbolic groups. In particular, it is proved that for every non-cyclic torsion-free hyperbolic group, there exists a non-Abelian torsion-free quotient group in which all proper subgroups are cyclic, and the intersection of any two of them is not trivial.