= Polynomial group growth from orbit packing
{title2=$|B_S(m)|\leq C(1+m)^n$}
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 $B(p,Dm+\varepsilon)$ and apply the <Bishop-Gromov inequality>. A finite <point stabilizer> contributes its order to the bound; freeness is unnecessary.
Back to article page
