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- Title
A geometric approach to the modified Milnor problem.
- Authors
Chen, Lina; Rong, Xiaochun; Xu, Shicheng
- 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 (n , d) > 0 such that if a compact Riemannian n -manifold M satisfies that Ricci curvature Ric M ≥ − (n − 1) , diameter d ≥ diam (M) and volume entropy h (M) < (n , d) , then the fundamental group π 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) < (n , d) , then h (M) = 0.
- Subjects
GEOMETRIC approach; RIEMANNIAN geometry; FINITE groups; GROUP theory; RIEMANNIAN manifolds; ENTROPY
- Publication
Communications in Contemporary Mathematics, 2024, Vol 26, Issue 5, p1
- ISSN
0219-1997
- Publication type
Article
- DOI
10.1142/S0219199723500189