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Home >> Seminar
 A Geometric Approach to the Modified Milnor Problem ³ÂÀöÄÈ ²©Ê¿(»ª¶«Ê¦·¶´óÑ§) Thursday, December 20th, 2018, 1:00 PM  ãÉÐÐÊýÑ§Â¥102±¨¸æÌü abstract£ºThe 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.

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