Structural results on the weak commutativity construction

Abstract The weak commutativity group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>χ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \chi(G) is generated by two isomorphic groups 𝐺 and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>G</m:mi> <m:mi>φ</m:mi> </m:msup> </m:math> G^{\varphi} subject to the relations <m:math xmlns:m="http://www.w3.org/1998/Math/Mat