Realizing residually finite groups as subgroups of branch groups

TL;DR

This paper demonstrates that any finitely generated, residually finite group can be embedded into a finitely generated perfect branch group, preserving key properties and enabling the construction of new groups with specific features.

Contribution

It introduces a method to embed residually finite groups into branch groups while maintaining properties like torsion and amenability, and constructs a non-amenable torsion branch group.

Findings

Embedding of residually finite groups into branch groups preserves key properties.

Construction of a finitely generated, non-amenable torsion branch group.

Demonstrates the versatility of branch groups in group theory.

Abstract

We prove that every finitely generated, residually finite group $G$ embeds into a finitely generated perfect branch group $\Gamma$ such that many properties of $G$ are preserved under this embedding. Among those are the properties of being torsion, being amenable, and not containing a non-abelian free group. As an application we construct a finitely generated, non-amenable torsion branch group.