Right-angled Artin groups and a generalized isomorphism problem for finitely generated subgroups of mapping class groups
Right-angled Artin groups and a generalized isomorphism problem for finitely generated subgroups of mapping class groups
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直角Artin群和映射类群的有限生成子群的广义同构问题
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发表时间：
2010
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通讯作者：
T. Koberda
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作者：
T. Koberda
考虑具有p个穿孔的亏格g的曲面图g，p的映射类群Modg，p，以及有限集合{f1，. . .，fk}的映射类，其中每一个要么是一个简单的封闭曲线的Dehn扭曲或伪Anosov同胚支持上连接的地下。本文证明了对所有充分大的N，映射类${{f_1^N，ldots，f_k ^N}}$$生成直角Artin群.它们生成的直角Artin群可以从映射类本身的组合拓扑中确定。当{f1，. . .，fk}是任意映射类，我们证明了这些映射类的足够大的幂会生成一个以受控方式嵌入直角Artin群的群。我们建立了真实的和复双曲流形的一些类似结果。我们还讨论了Modg，p的n-生成子群的同构问题的不可解性，并恢复了直角Artin群的同构问题是可解的这一事实.因此，我们刻画了许多自然发生的Modg，p的子群的同构类型。
Consider the mapping class group Modg,p of a surface Σg,p of genus g with p punctures, and a finite collection {f1, . . . , fk} of mapping classes, each of which is either a Dehn twist about a simple closed curve or a pseudo-Anosov homeomorphism supported on a connected subsurface. In this paper we prove that for all sufficiently large N, the mapping classes $${{f_1^N,ldots,f_k^N}}$$ generate a right-angled Artin group. The right-angled Artin group which they generate can be determined from the combinatorial topology of the mapping classes themselves. When {f1, . . . , fk} are arbitrary mapping classes, we show that sufficiently large powers of these mapping classes generate a group which embeds in a right-angled Artin group in a controlled way. We establish some analogous results for real and complex hyperbolic manifolds. We also discuss the unsolvability of the isomorphism problem for finitely generated subgroups of Modg,p, and recover the fact that the isomorphism problem for right-angled Artin groups is solvable. We thus characterize the isomorphism type of many naturally occurring subgroups of Modg,p.