A Geometric Approach to the Modified Milnor Problem

Abstract

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 $h(M)=0$.

25 pages

References in corpus (2)

• Quantitative Volume Space Form Rigidity Under Lower Ricci Curvature Bound
• Ricci curvature and isometric actions with scaling nonvanishing property

Cited by in corpus (1)

• On ends of finite-volume noncompact manifolds of nonpositive curvature