THE TOPOLOGY AND ANALYSIS OF THE HANNA NEUMANN CONJECTURE
THE TOPOLOGY AND ANALYSIS OF THE HANNA NEUMANN CONJECTURE
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DOI：
10.1142/s1793525311000611
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发表时间：
2011-11
影响因子：
0.8
通讯作者：
Igor Mineyev
中科院分区：
文献类型：
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作者：
Igor Mineyev
汉纳诺依曼猜想 (HNC) 的陈述是纯粹代数的：对于自由群 Γ 和 Γ 的任何非平凡有限生成子群 A 和 B，本文的目标是系统地开发允许 HNC 泛化的机制，并展示它们与拓扑和分析的关系。在拓扑方面，我们定义了复合体、树叶、复合体系统、花卉、花园以及图和表面的原子分解的沉浸。分析部分涉及使用希尔伯特模块的经典默里-冯诺依曼（！）维度。这也为强化汉纳诺依曼猜想（SHNC）及其推广提供了一种方法。我们提出了它的三个方面，分别称为正方形方法、对角线方法和排列方法。这三者中的每一个都来自系统的概念，并且每一个都引出了图和自由群之外的问题。给出了 SHNC 声明的部分结果、充分条件和概括。
The statement of the Hanna Neumann Conjecture (HNC) is purely algebraic: for a free group Γ and any nontrivial finitely generated subgroups A and B of Γ, The goal of this paper is to systematically develop machinery that would allow for generalizations of HNC and to exhibit their relations with topology and analysis. On the topological side we define immersions of complexes, leafages, systems of complexes, flowers, gardens, and atomic decompositions of graphs and surfaces. The analytic part involves working with the classical Murray–von Neumann (!) dimension of Hilbert modules. This also gives an approach to the Strengthened Hanna Neumann Conjecture (SHNC) and to its generalizations. We present three faces of it named, respectively, the square approach, the diagonal approach, and the arrangement approach. Each of the three comes from the notion of a system, and each leads to questions beyond graphs and free groups. Partial results, sufficient conditions, and generalizations of the statement of SHNC are presented.