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].
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