 Polynomial growth of properly discontinuous isometry groups

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.