Structural results on the weak commutativity construction

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> χ(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> Gφ subject to the relations <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mi>g</m:mi> <m:mo>,</m:mo> <m:msup> <m:mi>g</m:mi> <m:mi>φ</m:mi> </m:msup> <m:mo stretchy="false">]</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> [g,gφ]=1 for all <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>g</m:mi> <m:mo>∈</m:mo> <m:mi>G</m:mi> </m:mrow> </m:math> g∈ G . We obtain new expressions for the terms of the derived series and the lower central series of <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> χ(G) . We also present new bounds for the exponent of some sections of <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> χ(G) .