Research in Geometric Group Theory

Rutgers University New Brunswick, New Brunswick NJ

Investigators

Abstract

An exciting recent development in geometric group theory is Zlil Sela's beautiful work on the Tarski problem. This problem asks whether the elementary theory of a non-abelian free group is independent of the free group in question. Feighn proposes two projects in this direction, both joint with Mladen Bestvina. The first is a continuation of a project to simplify Sela's work. In the second, representing a new direction, a solution to a modified form of Tarski's problem is proposed. This will lead to new information about definable sets (objects of primary importance) and will answer questions of Rips and Sela. A third proposed project is to explore the extent to which the JSJ-decomposition of a group can be found algorithmically. The specific class of groups to be examined consists of graphs of free groups, which are of current interest on a number of fronts. With Guo-An Diao, the principal investigator has shown that the Grushko decomposition (a coarser decomposition than the JSJ) of these groups can be found algorithmically. There are two possible satisfactory resolutions, namely a construction of an algorithm to produce the JSJ-decomposition of a group in the class or a proof that this problem is unsolvable. For this class of groups, which exists on the boundary of what can be understood algorithmically, either resolution would be important. Geometric group theory is a relatively young branch of mathematics that uses geometric methods to understand groups. The idea to take a problem in group theory, translate it into a problem in geometry, and then use geometric methods to solve the translated problem. The principal investigator proposes to use geometric methods to explore two general areas. The first is motivated by Zlil Sela's geometric solution to the Tarski problem which on the face of it is a problem in the intersection of group theory and logic. Roughly, the problem is to distinguish groups using the truth of statements formed using only the usual symbols of logic ("there exists", "for all", variables, etc) and group operations. For example, you can tell if a group is abelian by asking whether it is true that, for all elements x and y, xy=yx. The first proposed project, joint with Mladen Bestvina, is to simplify and extend Sela's ideas. This project is ongoing and there has already been significant progress. The second proposed project follows another trend in group theory that is encapsulated by the question: How much group theory can a computer understand? Much as molecules can be decomposed into simpler pieces (atoms), groups can be often be decomposed into simpler groups. The project is to understand such decompositions for a class of groups called "graphs of free groups", a class which has attracted much recent attention. It turns out that decomposing groups translates into the geometric process of cutting a space into simpler pieces. The goal is to use this geometric idea to discover an algorithm that, given a group, finds its decomposition. The principal investigator, with Guo-An Diao, has already constructed an algorithm for a related process that can readily be implemented on a computer.