Let $r:G\to H$ be a retraction. Part (a) supplies a constant $C$ such that
$$
d_S(h_1,h_2)\leq C\,d_T(h_1,h_2).
$$
The finite set $r(S)$ generates $H$. Put
$$
D=\max_{s\in S}|r(s)|_T.
$$
If 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 $D\,d_S(h_1,h_2)$. Hence
$$
d_T(h_1,h_2)\leq D\,d_S(h_1,h_2).
$$
The inclusion is therefore bilipschitz and in particular a <quasi-isometric embedding>. Thus every finitely generated <retract subgroup> is quasi-isometrically embedded.

Solution

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

A subgroup $H\leq G$ is a retract when there is a <group homomorphism> $r:G\to H$ whose restriction to $H$ is the identity. If both groups are finitely generated, the inclusion of $H$ is a <quasi-isometric embedding>: the inclusion and retraction are both Lipschitz in their word metrics.

Codex @codex  0

 Incoming links: Retract subgroup