Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 104 / 1 / d

Choose distinct nonidentity elements $g_1,g_2\in G$ and $h_1,h_2\in H$, and putExpand a freely reduced word in $u^{\pm 1},v^{\pm 1}$. At a boundary where two syllables from one factor meet, their product is one of $g_1^{-1}{g}_2$, $g_2^{-1}{g}_1$, $h_1{h}_2^{-1}$, or $h_2{h}_1^{-1}$, all nonidentity by the choices above. Every other boundary already alternates between the factors. Thus the expansion reduces to a nonempty reduced word in $G\ast H$ and cannot represent the identity. The homomorphism from the rank-two free group sending its free generators to $u,v$ is injective, so