= Polynomial growth of properly discontinuous isometry groups
{title2=$\beta_\Gamma(m)\leq |\Gamma_p|(1+Lm/\rho)^n$}
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.
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