Splittings of Groups

DMS-0204509Mark E. Feightenly提出的工作是在该地区的几何群论。一个激励性的问题是：给定一个群G的图，在顶点和边群上有某种限制，关于G可以说什么？ 例如，G是一个非平凡的自由积吗？G在整数上分裂吗？什么是G的JSF分解？一个重点将是有限的图形的非线性生成的自由群。作为沿着这些路线取得进展的一个标志，主要研究者与他的研究生刁国安一起发现了一种算法，用于确定一个有限图的非平凡生成的自由群是否是一个非平凡的自由积。拓扑空间通常通过将它们切开，然后考虑得到的更简单的部分来分析。一个简单的例子是，如果一个圆被切割，那么一个弧仍然存在。类似地，群的研究通常是沿着沿着子群切开。由于群可以表示为空间的对称性，群的分裂和空间的分裂是同一构造的两种表现。例如，上面的圆的分裂给出了整数的描述。 其中最令人兴奋的发展inggrouptheory在过去的20年里是描述的里普斯-塞拉的所有分裂的群体超过子群，如整数。Dunwoody-Sageev和Fujiwara-Papasoglu扩展了可以描述的分裂类型。主要研究者将探索这些描述可以在算法上进行的程度。

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.