Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 19 / 5 / a / Solution

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.