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arXiv 1607.03043math.GRmath.GT
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自由群的对称自同构、BNSR-不变量和有限性

Symmetric automorphisms of free groups, BNSR-invariants, and finiteness properties

Matthew C. B. Zaremsky

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25 pages, 3 figures
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摘要：

群$G$的BNSR-不变量是一个几何不变量序列$Σ^1(G)\supseteq Σ^2(G) \supseteq \cdots$，它揭示了$G$的某些子群的有限性的重要信息。考虑自由群$F_n$的对称自同构群$ΣAut_n$和纯对称自同构群$PΣAut_n$，考察它们的BNSR-不变量.我们证明了对于$n\ge 2$，$PΣAut_n$的所有"正“和”负“字符类都位于$Σ^{n-2}(PΣAut_n)\setminus Σ^{n-1}(PΣAut_n)$中.我们用它来证明对于$n\ge 2$，$Σ^{n-2}(ΣAut_n)$等于$ΣAut_n$的全特征球$S^0$，但$Σ^{n-1}(ΣAut_n)$是空的，所以特别是换位子群$ΣAut_n'$是$F_{n-2}$类型，但不是$F_{n-1}$。我们的技术涉及应用莫尔斯理论的复杂的对称标记仙人掌图。

英文摘要：

The BNSR-invariants of a group $G$ are a sequence $Σ^1(G)\supseteq Σ^2(G) \supseteq \cdots$ of geometric invariants that reveal important information about finiteness properties of certain subgroups of $G$. We consider the symmetric automorphism group $ΣAut_n$ and pure symmetric automorphism group $PΣAut_n$ of the free group $F_n$, and inspect their BNSR-invariants. We prove that for $n\ge 2$, all the ``positive'' and ``negative'' character classes of $PΣAut_n$ lie in $Σ^{n-2}(PΣAut_n)\setminus Σ^{n-1}(PΣAut_n)$. We use this to prove that for $n\ge 2$, $Σ^{n-2}(ΣAut_n)$ equals the full character sphere $S^0$ of $ΣAut_n$ but $Σ^{n-1}(ΣAut_n)$ is empty, so in particular the commutator subgroup $ΣAut_n'$ is of type $F_{n-2}$ but not $F_{n-1}$. Our techniques involve applying Morse theory to the complex of symmetric marked cactus graphs.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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