A Geometric Approach to the Modified Milnor Problem

The Milnor Problem (modified) in the theory of group growth asks whether any finite presented vanishing algebraic entropy has at most polynomial growth. We show that a positive answer to is equivalent Nilpotency Conjecture Riemannian geometry: given $n, d>0$, there exists constant $\epsilon(n,d)>0$ such if compact $n$-manifold $M$ satisfies Ricci curvature $\op{Ric}_M\ge -(n-1)$, diameter $d\ge \op{diam}(M)$ and volume $h(M)<\epsilon(n,d)$, then fundamental $\pi_1(M)$ virtually nilpotent. will verify some cases, we gap phenomena for more cases i.e., $h(M)=0$.