Polynomial group growth from orbit packing ($|B_S(m)|\le C(1+m{)}^n$)

A properly discontinuous group action by isometries of a finitely generated group on a complete $n$-dimensional Riemannian manifold of nonnegative Ricci curvature yields the displayed group growth bound. Distinct orbit points are uniformly separated, while a word of length $m$ moves a basepoint by at most $Dm$. Pack disjoint equal-volume small balls about the orbit points into $B(p,Dm+\epsilon )$ and apply the Bishop-Gromov inequality. A finite point stabilizer contributes its order to the bound; freeness is unnecessary.