Splittings of Groups

DMS-0204509马克·E·费根建议的工作是在几何群论领域。一个激动人心的问题是：给出一个群G的图，对顶点和边群有某种限制，关于G可以说些什么？例如，G是否是非平凡的自由积？G是否在整数上拆分？G的JSJ分解是什么？重点放在有限生成自由群的有限图上。作为在这些方面取得进展的一个标志，首席研究员和他的研究生刁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的自由积。分析拓扑空间的方法通常是将它们剖开，然后考虑得到的更简单的片断。一个简单的例子是，如果一个圆被割断，那么圆弧就会保留下来。类似地，通常通过从子群切开群来研究群。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结构的两种表现形式。例如，上面圆圈的拆分提供了对整数的描述。在过去的20年里，群论最激动人心的发展之一是Rips-Sela刻画了群在子群(如整数)上的所有分裂。Dunwoody-Sageev和Fujiwara-Papasoglu扩展了可以描述的分裂类型。首席调查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.