Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 133 / 3 / b / Solution

Let $r:G\to H$ be a retraction. Part (a) supplies a constant $C$ such thatThe finite set $r(S)$ generates $H$. PutIf a shortest $S$-word represents $h_1^{-1}{h}_2\in H$, applying the group homomorphism $r$ gives a $T$-word for the same element of length at most $D{d}_S(h_1,h_2)$. HenceThe inclusion is therefore bilipschitz and in particular a quasi-isometric embedding. Thus every finitely generated retract subgroup is quasi-isometrically embedded.