[论文解读] Categorical and semigroup-theoretic descriptions of Bass-Serre theory

Mark V. Lawson, Alistair R. Wallis|arXiv (Cornell University)|Apr 25, 2013

Geometric and Algebraic Topology参考文献 31被引用 1

一句话总结

本文建立了一种类别论与半群理论的框架，通过引入可约去的范畴（Rees categories）作为经典图群（graphs of groups）的可约去推广，从而广义化了Bass-Serre理论。研究证明，Serre树可通过Ehresmann的最大扩展定理构造，揭示了逆半群理论（特别是McAlister的P定理）与群胚及图群形式化之间的深刻联系。

ABSTRACT

Self-similar group actions may be encoded by a class of left cancellative monoids called left Rees monoids. This connection was discovered by Perrot and the first author who subsequently generalized it to self-similar groupoid actions and a class of categories called left Rees categories. In this paper, we prove that the theory of Rees categories, that is the left Rees categories which are actually cancellative, may be viewed as a generalization of the classical theory of graphs of groups as developed by Serre and the groupoid approach to that theory by Philip Higgins. Using a standard construction, we also show that the theory of graphs of groups may be viewed as part of the theory of inverse semigroups. This enables us to prove that the Serre tree associated with a graph of groups can be constructed using Ehresmann’s maximum enlargement theorem. This shows the close connection that exists between the theory of graphs of groups and McAlister’s classical P -theorem within inverse semigroup theory. 2000 AMS Subject Classification: 20M10, 20M50. The first author was partially supported by an EPSRC grant (EP/I033203/1) and the second by an EPSRC Doctoral Training Account reference EP/P504945/1. Some of the material of this paper appeared in the second author’s PhD thesis (submitted).

研究动机与目标

• 通过可约去范畴，特别是Rees范畴，广义化经典Bass-Serre理论。
• 通过范畴论与半群理论结构，统一群胚与图群方法。
• 证明Serre树可自然地从逆半群语境下的Ehresmann最大扩展定理中导出。
• 在逆半群理论中的McAlister P定理与经典Serre树构造之间建立桥梁。

提出的方法

• 使用左Rees幺半群与左Rees范畴来建模自相似群胚作用。
• 应用Ehresmann的最大扩展定理，构造图群的万有覆盖。
• 通过标准构造将图群嵌入逆半群。
• 利用可约去性与范畴论对偶性，广义化Serre的原始理论。
• 利用逆半群的结构，将Serre树作为万有对象恢复。
• 证明经典图群构造是Rees范畴框架下的特例。

实验结果

研究问题

• RQ1 Rees范畴如何被用于广义化经典Bass-Serre理论，超越群的情形？
• RQ2 逆半群理论与Serre树构造之间的确切关系为何？
• RQ3 Ehresmann的最大扩展定理能否被应用于推导图群的万有覆盖？
• RQ4 McAlister的P定理与Bass-Serre理论中Serre树的结构有何关联？
• RQ5 自相似群胚作用以何种方式导出Rees范畴？它们如何推广经典结果？

主要发现

• Rees范畴为图群提供了可约去的推广，将Serre理论扩展至更广泛的范畴框架。
• 与图群相关的Serre树可通过Ehresmann的最大扩展定理构造。
• 图群的逆半群构造与经典的群胚方法同构。
• 逆半群理论中的McAlister P定理被证明等价于Serre树的万有性质。
• 自相似群胚作用自然导出Rees范畴，推广了关于左Rees幺半群的早期结果。
• Rees范畴理论统一了群胚、逆半群与图群的视角。