Two properties of the filtration of the outer automorphism groups of certain groups
Two properties of the filtration of the outer automorphism groups of certain groups
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某些群的外自同构群的过滤的两个性质
DOI：
10.1007/bf02571892
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发表时间：
1995
影响因子：
0.8
通讯作者：
Mamoru Asada
中科院分区：
文献类型：
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作者：
Mamoru Asada
设G是G生成的离散群或亲群，f是固定素数.设F表示G的外自同构群。群F有一个过滤{F [rrt]} m> 0，它是由G的降序中心列自然导出的。(For{ff [m]} m> O的定义，见第1节。）本文的目的是证明在G上的某些假设下，交换群F [m]/F [m+ 1]（m> 1）是有限秩的自由Z（或Zi）-模，且交[”lm~= oF [m]约化为{1}（在pro-情形下）.如果G是亏格g> 2的曲面群的pro-f完备化，则上述假设成立。因此，我们的结果补充了Asada和Kaneko [3]的工作，其中我们计算了n-生成Z-模Fg [rn]/Fg [m+ 1]（m> 1）的自由部分的秩，作为研究Fg中Galois表示的一个必要条件.这里，Fy是g~）的外自同构群。
Let G be a finitely generated discrete or pro-(group, f being a fixed prime number. Let F denote the outer automorphism group of G. The group F has a filtration {F [rrt]} m> 0 which is naturally induced from the descending central series of G.(For the definition of {ff [m]} m> O, see Sect. 1.) Our aim in this paper is to show, under some assumptions on G, that the abelian group F [m]/F [m+ 1](m> 1) is a free Z (or Zi)-module of finite rank and that the intersection [" lm~= oF [m] reduces to {1}(in the pro-(case). The assumptions are satisfied if G is the pro-f completion~) of the surface group of genus g> 2. Hence our results supplement our previous work Asada and Kaneko [3], in which we have calculated the rank of the free part of the finitely generated Z/-module Fg [rn]/Fg [m+ 1](m> 1) as preliminaries for the investigation on the Galois representation in Fg. Here, Fy is the outer automorphism group of g~).