Splittings of Groups

mark E. feighn提议的工作是在几何群论领域。一个激励问题是：给定一个群G的图，在顶点群和边群上有某种限制，关于G我们能说些什么？例如，G是否为非平凡自由积？；G会在整数上分裂吗？G的jsj分解是什么？重点将放在有限生成自由群的有限图上。作为这方面进展的一个标志，首席研究员和他的研究生王国安发现了一种算法，用于确定有限生成的自由群的有限图是否是非平凡自由积。拓扑空间的分析通常是将其切开，然后考虑得到的更简单的部分。一个简单的例子是，如果一个圆被切割，那么弧仍然存在。类似地，群体的研究通常是沿着子群体切开。由于群可以被表示为空间的对称，群的分裂和空间的分裂是同一结构的两种表现形式。例如，上面的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瑞普斯-塞拉对群在子群（如整数）上的所有分裂的描述。Dunwoody-Sageev和Fujiwara-Papasoglu扩展了可以描述的分裂类型。首席研究员将探索这些描述在多大程度上可以通过算法实现。

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.