题目：Nilpotency Conjecture, a Geometric Approach to Milnor's Problem on Group Growth
摘要：This is a joint work with Lina Chen and Prof. Xiaochun Rong. 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. It is conjecture by Grigorchuk and Park to be true. We show that a positive answer to the modified Milnor Problem 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 >=-(n-1), diameter <=d and volume entropy <\epsilon(n,d), then the fundamental group of M has a nilpotent subgroup of finite index. It is our hope that this equivalence will bring geometric tools into the study of Milnor Problem, since by the equivalence progresses made in either problem will shed a light on the other.
From the geometric viewpoint, Nilpotency Conjecture is a natural extension of earlier known results and conjectures on the fundamental group of manifolds under lower curvature bounds. 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 $h(M)=0$. The gap phenomena was raised to the author by S. Honda in 2016.
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