On weak commutativity in groups

For a group G   we study homological and homotopical properties of the group χ(G)=〈G,Gψ|[g,gψ]=1 for g∈G〉χ(G)=〈G,Gψ|[g,gψ]=1 for g∈G〉. In particular, we show that the operator χ   preserves the soluble of type FP∞FP∞ property.