Splittings of Groups

DMS-0204509 Mark E. Feighn The 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 in group 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.