# Groups with finitely many conjugacy classes and their automorphisms

## TL;DR

This paper constructs finitely generated torsion-free groups with finitely many conjugacy classes using advanced combinatorial and small cancellation techniques, and explores their automorphism groups and embeddings.

## Contribution

It introduces new methods combining classical and modern group theory to build groups with specific conjugacy and automorphism properties, including realization of any countable group as an outer automorphism group.

## Key findings

- Constructed finitely generated torsion-free groups with finitely many conjugacy classes.
- Proved that any countable group can be realized as an outer automorphism group of a specific finitely generated group.
- Developed new embedding results for such groups.

## Abstract

We combine classical methods of combinatorial group theory with the theory of small cancellations over relatively hyperbolic groups to construct finitely generated torsion-free groups that have only finitely many classes of conjugate elements. Moreover, we present several results concerning embeddings into such groups. As another application of these techniques, we prove that every countable group $C$ can be realized as a group of outer automorphisms of a group $N$, where $N$ is a finitely generated group having Kazhdan's property (T) and containing exactly two conjugacy classes.