A finitely generated branch group of exponential growth without free subgroups

TL;DR

This paper constructs a specific branch group with exponential growth that lacks free subgroups, answering a longstanding question and revealing new properties about its subgroup structure and growth behavior.

Contribution

It introduces a new example of a branch group with exponential growth that contains no non-abelian free subgroups, expanding understanding of group growth and subgroup dynamics.

Findings

Group has exponential growth without free subgroups

Every normal subgroup is finitely generated

Proper quotients are soluble

Abstract

We will give an example of a branch group $G$ that has exponential growth but does not contain any non-abelian free subgroups. This answers question 16 from \cite{Bartholdi} positively. The proof demonstrates how to construct a non-trivial word $w_{a,b}(x,y)$ for any $a,b \in G$ such that $w_{a,b}(a,b) = 1$. The group $G$ is not just-infinite. We prove that every normal subgroup of $G$ is finitely generated as an abstract group and every proper quotient soluble. Further, $G$ has infinite virtual first Betti number but is not large.