A finitely generated branch group of exponential growth without free subgroups

2012/07/27 by Elisabeth Fink, Fink, Elisabeth
Mathematics · #FOS: Mathematics #Group Theory (math.GR) #math.GR

paper · pdf · doi:10.48550/arxiv.1207.6548

16 pages, 1 figure

arxiv created 2012/09/24 · arxiv updated 2012/09/25

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 \citeBartholdi positively. The proof demonstrates how to construct a non-trivial word w_a,b(x,y) for any a,b ∈ 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.

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