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 [Formula: see text], there exists a constant [Formula: see text] such that if a compact Riemannian [Formula: see text]-manifold [Formula: see text] satisfies that Ricci curvature [Formula: see text], diameter [Formula: see text] and volume entropy [Formula: see text], then the fundamental group [Formula: see text] 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 [Formula: see text], then [Formula: see text].