A Geometric Approach to the Modified Milnor Problem

发布时间：2019-04-17 作者：浏览次数：

Speaker:	陈丽娜	DateTime:	2019年4月24号（周三）下午4:00-5:00
Brief Introduction to Speaker: 陈丽娜，华东师范大学博士。		Place:	六号楼二楼报告厅
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 $h(M)="0$." This is a joint work with Professor Xiaochun Rong and Shicheng Xu.