Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 149 / 3 / c

Write $N=[G,G]$ and identify it with $(\mathbb{C},+)$ as in part (a). If every two members of $A$ commuted, then $⟨A⟩$ would be Abelian, contrary to hypothesis. Thus some commutator of two members of $A$ is a nonidentity translation in $A^4\cap N$. Consequently the translation-coordinate setcontains both zero and a nonzero element.

The identityshows that $T\pi (A)$ is contained in the translation coordinates of $A^6\cap N$, while $T+T$ is contained in those of $A^8\cap N$. The intersection of an approximate group power with a subgroup therefore givesApply the Solymosi sum-product theorem over the complex numbers with $U=V=T$ and $W=\pi (A)$. Since $1\in \pi (A)$, its hypotheses hold, andCancelling $|T{|}^2$ and absorbing the absolute constant proves