Weak commutativity and finiteness properties of groups

Abstract

We consider the group $\mathfrak{X}(G)$ obtained from $G\ast G$ by forcing each element $g$ in the first free factor to commute with the copy of $g$ in the second free factor. Deceptively complicated finitely presented groups arise from this construction: $\mathfrak{X}(G)$ is finitely presented if and only if $G$ is finitely presented, but if $F$ is a non-abelian free group of finite rank then $\mathfrak{X}(F)$ has a subgroup of finite index whose third homology is not finitely generated.

12 pages, 2 figures. Final version to appear in Bulletin of the London Math Soc

References in corpus (1)

• Uncountably many groups of type $FP$

Cited by in corpus (2)

• The weak commutativity construction for Lie algebras
• On the exponent of the Weak commutativity group $\chi (G)$