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