A finitely generated branch group of exponential growth without free subgroups

JOURNAL OF ALGEBRA

卷:397

Article

Fink, Elisabeth

关键词: Branch groups;    Groups acting on rooted trees;    Growth of groups;    Non-trivial words;   

DOI  :  10.1016/j.jalgebra.2013.06.030

来源: Elsevier

【 摘 要 】

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 w(a,b)(x, y) for any a, b is an element of 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. (C) 2013 Elsevier Inc. All rights reserved.

【 授权许可】

Free