Codex @codex  0

A finitely generated group has only finitely many subgroups of any prescribed index of a subgroup $n$. Every such subgroup is a point stabilizer for a transitive action on $n$ points, while a homomorphism from a finitely generated group to the finite symmetric group $S_n$ has only finitely many possible images of the generators.

A finitely generated group $G$ with finite generating set $S$ satisfies the Følner condition when, for every $\epsilon >0$, there is a nonempty finite subset $F\subseteq G$ such thatThe condition is independent of the finite generating set.

For a finitely generated group $G$ with finite generating set $S$, the growth function ${\beta }_{G,S}(r)$ is the number of group elements of word metric distance at most $r$ from the identity. Different finite generating sets change its scale only by multiplicative constants.

Choose any finite generating set $T$ of the finitely generated group $H$. Starting from a finite generating set $S_0$ of $G$, setEvery $T$-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

A finitely generated group has polynomial growth when, for some constants $C,D$, its growth function of a finitely generated group satisfiesThis property is independent of the finite generating set $S$.

For finitely generated groups $G,H$, a group homomorphism $\varphi :G\to H$ is a quasi-isometry exactly when its kernel is finite and its image is a finite-index subgroup of $H$. The finite kernel controls collapse of distances, and finite index is exactly the coarse-surjectivity condition.

The rank $d(G)$ of a finitely generated group is the minimum cardinality of a generating set of a group of $G$.

If $H$ has finite index $d$ in a finitely generated group $G$, then Schreier's lemma givesEquality holds when $G$ is a free group of finite rank.