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Every finitely generated group of polynomial growth of a group has a virtually nilpotent group structure. Applying the one-scale virtual nilpotence theorem gives a finite normal subgroup with nilpotent quotient; centralizing that finite subgroup produces a nilpotent subgroup of finite index.

Choose a finite symmetric generating set $S$ for $\Gamma$, a base point $p$, and put $L={\max }_{s\in S}d(p,sp)$. If the word length of $\gamma$ is at most $m$, the triangle inequality and invariance of Riemannian distance under isometries give $d(p,\gamma p)\le Lm$.

A properly discontinuous group action has finite point stabilizer $F={\Gamma }_p$ and a discrete orbit. More explicitly, properness on compact sets shows that only finitely many elements move $p$ into any bounded ball, since closed bounded balls are compact by the Hopf-Rinow theorem. Thus there is $\rho >0$ such that distinct orbit points have distance more than $2\rho$: choose it using the finitely many orbit points within a fixed small ball of $p$, and translate the resulting separation by isometries. If the whole orbit is one point, the group itself is finite and the conclusion is immediate.

Let $N_m$ be the number of distinct orbit points represented by the word metric ball of radius $m$. Their radius-$\rho$ balls are disjoint, have equal positive volume $v_0=\mathrm{Vol}B(p,\rho )$, and lie inside $B(p,Lm+\rho )$. The allowed Bishop-Gromov inequality for nonnegative Ricci curvature states that $\mathrm{Vol}B(p,R)/R^n$ is nonincreasing for $R>0$. ThereforeEach orbit point has at most $|F|$ preimages inside the word ball, since its full fiber is a coset of $F$. For the growth function of a finitely generated group, this givesHence $\Gamma$ has polynomial growth of a group of degree at most $n$. The finite stabilizer factor makes the proof valid even when the properly discontinuous action is not free. This is polynomial growth of properly discontinuous isometry groups.

Choose a finite symmetric generating set $S$ for $\Gamma$, let $B_S(m)$ be its word metric ball, and fix $p\in M$. Set $D={\max }_{s\in S}d(p,sp)$. The triangle inequality and invariance of distance implyThe properly discontinuous group action has a finite point stabilizer $H={\Gamma }_p$. It also gives uniform separation of distinct orbit points. To see this, only finitely many group elements can send $p$ into a fixed compact ball about $p$, by proper discontinuity and the Hopf-Rinow theorem. The positive distances among this finite list, together with the radius of the ball for all other elements, have a positive lower bound $\delta$. Thus $d(\gamma p,{\gamma }^{\prime }p)\ge \delta$ whenever the centers differ.

Take $0<\epsilon <\delta /2$ and write $v=\mathrm{Vol}B(p,\epsilon )>0$. Balls of radius $\epsilon$ about distinct orbit points are disjoint and all have volume $v$, because the action is by isometries. Those corresponding to words of length at most $m$ lie in $B(p,Dm+\epsilon )$. Each orbit point arises from at most $|H|$ group elements in $B_S(m)$, soThe final inequality is the allowed Bishop-Gromov inequality for nonnegative Ricci curvature. If the entire group fixes $p$, proper discontinuity already makes it finite and the conclusion is immediate. Otherwise the preceding separation argument applies. ThereforeThis is polynomial growth of a group of degree at most $n$, proved by polynomial group growth from orbit packing. Neither freeness of the action nor compactness of the quotient was assumed.

A finitely generated group has polynomial growth of a group if its growth function is bounded above by $C(1+r{)}^D$ for some constants $C,D$.

Let $G$ be generated by $s_1,\dots ,s_k$. Since $G$ is a two-step nilpotent group, every group commutator $[s_i,s_j]$ is central. Commuting letters past one another therefore collects every word of length at most $n$ into the formThe generator exponents satisfy $|a_i|\le n$. At most $n(n-1)/2$ exchanges are needed during collection, so $|b_{ij}|\le n^2$ is a sufficient bound. With $m=k(k-1)/2$, the number of possible collected expressions is at mostRelations can only reduce this count. This proves the polynomial growth of a finitely generated two-step nilpotent group, so every finitely generated two-step nilpotent group has polynomial growth.

Every finitely generated two-step nilpotent group has polynomial growth of a group. If $s_1,\dots ,s_k$ generate it, centrality of the commutators collects every word of length at most $n$ intowhere $|a_i|\le n$ and $|b_{ij}|=O(n^2)$. There are therefore only polynomially many possible collected words.

A finitely generated group of isometries acting properly discontinuously on a complete connected $n$-dimensional Riemannian manifold with nonnegative Ricci curvature has polynomial growth of a group of degree at most $n$. Choose $\rho >0$ so distinct points of an orbit are separated by more than $2\rho$. Let $L$ be the largest displacement of the base point by a generator. The radius-$m$ word metric ball maps into the metric ball of radius $Lm$, with fibers of size the finite stabilizer $|{\Gamma }_p|$. Disjoint radius-$\rho$ balls around these orbit points have equal volume and fit inside the ball of radius $Lm+\rho$. The Bishop-Gromov inequality bounds their number by $(1+Lm/\rho {)}^n$, giving the displayed estimate.