On weak commutativity in groups

Abstract

For a group G we study homological and homotopical properties of the group $\chi \left(G\right)=〈G,G^{\psi }\left|\right[g,g^{\psi }]=1\text{ for }g\in G〉$. In particular, we show that the operator χ preserves the soluble of type $F{P}_{\infty }$ property.