Choose any finite generating set $T$ of the <finitely generated group> $H$. Starting from a finite generating set $S_0$ of $G$, set
$$
S=S_0\cup T.
$$
Every $T$-word in $H$ of length at most $r$ is also an $S$-word in $G$ of length at most $r$. The inclusion of the corresponding word-metric balls is injective, so the <growth function of a finitely generated group> satisfies
$$
\boxed{\beta_{H,T}(r)\leq\beta_{G,S}(r)\qquad(r\geq0).}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 143 / 3 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-143/3/b/solution