A note on curvature and fundamental group
A note on curvature and fundamental group
复制标题
DOI：
10.4310/jdg/1214501132
复制
发表时间：
1968
影响因子：
2.5
通讯作者：
J. Milnor
中科院分区：
文献类型：
--
作者：
J. Milnor
定义与一个随机生成的群相关联的增长函数γ，并为该群指定生成元{gl 7-，gp}的选择，如下所示（比较[9]）。对于每个正整数s，设γ（s）是在指定的生成元及其逆中可以表示为长度< s的字的不同群元素的个数。(For例如，如果群是秩为2的自由阿贝尔群，具有指定的生成元x和y，则γ（s）= 2s +2s-f 1。我们将看到，γ（s）作为s -· oo的渐近行为在一定程度上与生成元的特定选择无关（引理1）。本注记将利用由R引起的曲率和体积的不等式。L. Bishop [1]、[2]和P.G.G. Nther [3]等，证明了两个定理。定理1.如果M是一个完备的n维黎曼流形，其平均曲率张量R^处处半正定，则与基本群7ΓjM的任意n维生成子群相关联的增长函数γ（s）必须满足
Define the growth function γ associated with a finitely generated group and a specified choice of generators {gl7 -, gp} for the group as follows (compare [9]). For each positive integer s let γ(s) be the number of distinct group elements which can be expressed as words of length < s in the specified generators and their inverses. (For example, if the group is free abelian of rank 2 with specified generators x and y, then γ(s) = 2s + 2s -f 1.) We will see that the asympotic behavior of γ(s) as s —• oo is, to a certain extent, independent of the particular choice of generators (Lemma 1). This note will make use of inequalities relating curvature and volume, due to R. L. Bishop [1], [2] and P. Gύnther [3], to prove two theorems. Theorem 1. // M is a complete n-dimensional Riemannian manifold whose mean curvature tensor R^ is everywhere positive semidefinite, then the growth function γ(s) associated with any finitely generated subgroup of the fundamental group 7ΓjM must satisfy