Splittings of Groups

DMS-0204509Mark E. Feighn 提出的工作属于几何群论领域。一个激励性的问题是：给定一个群 G 的图，对顶点和边群有某种限制，关于 G 可以说什么呢？ 例如，G 是一个不平凡的免费产品吗？ G 是否分裂整数？；什么是 G 的 JSJ 分解？重点将放在有限生成的自由群的有限图上。作为沿着这些思路取得进展的标志，首席研究员与他的研究生刁国安找到了一种算法，用于确定有限生成的自由群的有限图是否是一个非平凡的自由产品。拓扑空间通常通过将其切开然后考虑所得的更简单的部分来分析。一个简单的例子是，如果切割一个圆，则留下一条弧。类似地，通常通过沿着子群将群切开来研究群。由于群可以表示为空间的对称性，因此群的分裂和空间的分裂是同一构造的两种表现形式。例如，上面圆的分裂给出了整数的描述。 过去 20 年群论中最令人兴奋的发展之一是 Rips-Sela 对群在子群（例如整数）上的所有分裂的描述。 Dunwoody-Sageev 和 Fujiwara-Papasoglu 扩展了可描述的分裂类型。首席研究员将探索这些描述可以在多大程度上通过算法进行。

DMS-0204509Mark E. FeighnThe proposed work is in the area of geometric group theory. A motivating question is: Given a graph of groups G with some kind of restriction on the vertex and edge groups, what can be said about G? For example, is G a non-trivial free product?; does G split over the integers?; what is the JSJ-decomposition of G? A focus will be on finite graphs of finitely generated free groups. As an indication of progress along these lines, with his graduate student Guo-An Diao, the principal investigator has found an algorithm for determining whether a finite graph of finitely generated free groups is a non-trivial free product.Topological spaces are often analyzed by cutting them open and then considering the resulting simpler pieces. An easy example of this is that if a circle is cut then an arc remains. Analogously, groups are often studied by cutting them open along subgroups. Since groups may be represented as symmetries of spaces, splittings of groups and splittings of spaces are two manifestations of the same construction. For example, the splitting of the circle above gives a description of the integers. One of the most exciting developments ingroup theory over the past 20 years is the description by Rips-Sela of all splittings of groups over subgroups such as the integers. Dunwoody-Sageev and Fujiwara-Papasoglu have extended the types of splittings that can be described. The principal investigator will explore the extent to which these descriptions can be made algorithmically.