On a problem of Magnus
On a problem of Magnus
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关于马格努斯问题
DOI：
10.1016/0021-8693(82)90320-9
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发表时间：
1982
影响因子：
0.9
通讯作者：
D. Johnson
中科院分区：
文献类型：
--
作者：
D. Johnson
如果j= i+1，则（1）= a，i，3否则，由此保持（ujm，1< j，< n}，m> 2）在F中的正常闭包N。众所周知[31]，这种作用是忠实的; B对F，，/N的诱导作用也是忠实的，这构成了Birman和希尔登[21]的定理7。Magnus [S]在证明B嵌入某个多项式环的自同构群中时，定理4.11使用了Birman-Hilden定理的m= 2的情形，并指出迄今为止还没有找到这一结果的代数证明，本文的目的是通过在一个特殊情形下证明一个更强的结果来弥补这一不足，至少部分地弥补这一不足。即，当m为偶数时，B，忠实地作用于F，IN，.
= CTi,, a, aj,,, if j= i+ l(1)= a, i 3 otherwise, whereby the normal closure N, in F, of (ujm 1 1< j,< n}, m> 2, is preserved. It is well known [31 that this action is faithful; that the induced action of B, on F,,/N, is also faithful constitutes Theorem 7 of Birman and Hilden [21. In proving that B, embeds in the automorphism group of a certain polynomial ring, Magnus [S, Theorem 4.11 uses the case m= 2 of the Birman-Hilden theorem, and makes the comment that no algebraic proof of this result hs so far been found. The aim of this article is to remedy this deficiency, at least in part, by proving a stronger result in a special case. namely, that when m is even, B, acts faithful@ on the commutator subgroup of F,, IN,.