$\mathcal{K}_{g}$ is not finitely generated
$\mathcal{K}_{g}$
is not finitely generated
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$mathcal{K}_{g}$ 不是有限生成的
DOI：
10.1007/s00222-009-0202-x
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发表时间：
2006
影响因子：
3.1
通讯作者：
Benson Farb
中科院分区：
文献类型：
--
作者：
Daniel K. Biss;Benson Farb
亏格为g的闭可定向曲面的映射类群的两个自然子群是Torelli群Ug，由诱导曲面同调的映射类组成，以及它的子群% g，由Dehn twists关于分离曲面的简单闭曲线生成。关于这些群的研究已经取得了许多成果。特别地，J. Powell [Proc.Amer.Math.Soc.68（1978），no.3，347-350; MR 0494115（58#13045）]发现了Ug的生成元;在g= 2的情况下，它们正好是关于分离曲线的Dehn扭曲，因此% 2= U2。A.米勒和评论家[拓扑应用22（1986），no. 1，43-49; MR 0831180（87 h：57015）]证明了U 2和因此% 2不是双生成的。在[发明。163（2006），no. 1，213-226; MR 2208422（2006 m：57025）]，作者扩展了米勒和评论者的方法，给出了对于所有g≥ 2，% g不是均生成的论点。在这个勘误表中，他们报告说他们的扩展包含一个致命错误。他们认为佐藤正俊（Masatoshi Sato）的发现，以及汤姆·丘奇（Tom Church）构建了一个明确的例子。g= 2的情况不受影响，但g≥ 3的有限生成问题应该再次被认为是开放的。
Two natural subgroups of the mapping class group of a closed orientable surface of genus g are the Torelli group Ug, consisting of the mapping classes that induce the identity on the homology of the surface, and its subgroup% g, generated by Dehn twists about simple closed curves that separate the surface. Many results concerning these groups have been obtained. In particular, J. Powell [Proc. Amer. Math. Soc. 68 (1978), no. 3, 347–350; MR0494115 (58# 13045)] found generators for Ug; in the case g= 2 they are exactly the Dehn twists about separating curves, so% 2= U2. A. Miller and the reviewer [Topology Appl. 22 (1986), no. 1, 43–49; MR0831180 (87h: 57015)] proved that U2 and hence% 2 are not finitely generated. In [Invent. Math. 163 (2006), no. 1, 213–226; MR2208422 (2006m: 57025)] the authors extended the approach of Miller and the reviewer to give an argument that% g is not finitely generated for all g≥ 2. In this erratum they report that their extension contains a fatal error. They credit Masatoshi Sato with its discovery, and Tom Church with the construction of a clarifying example. The case of g= 2 is unaffected, but the problem of finite generation for g≥ 3 should again be considered open.