On self-similarity of wreath products of abelian groups

TL;DR

This paper investigates the properties of self-similar wreath products of abelian groups, establishing conditions under which these groups have finite rank or torsion properties, and providing explicit examples.

Contribution

It proves that self-similar free abelian groups have finite rank and characterizes torsion properties in self-similar wreath products of abelian groups.

Findings

Self-similar free abelian groups have finite rank.

If X is torsion-free, then B has finite exponent.

Constructed a self-similar group with infinite rank abelian B.

Abstract

We prove that a self-similar free abelian group has finite rank. We apply the result to self-similar wreath products of abelian groups $G=BwrX$. We show that if $X$ is torsion-free, then $B$ is torsion of finite exponent. Furthemore, we construct a self-similar group $G=BwrC_{2}$ where $B$ is free abelian of infinite rank.