A finitely generated branch group of exponential growth without free subgroups

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 Bartholdi et al. (2003) [1] positively. The proof demonstrates how to construct a non-trivial word wa,b(x,y)wa,b(x,y) for any a,b∈Ga,b∈G such that wa,b(a,b)=1wa,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.