Research in Geometric Group Theory

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An exciting recent development in geometric group theory is ZlilSela's beautiful work on the Tarski problem. This problem asks whetherthe elementary theory of a non-abelian free group is independent ofthe free group in question. Feighn proposes two projects in thisdirection, both joint with Mladen Bestvina. The first is acontinuation of a project to simplify Sela's work. In the second,representing a new direction, a solution to a modified form ofTarski's problem is proposed. This will lead to new information aboutdefinable sets (objects of primary importance) and will answerquestions of Rips and Sela. A third proposed project is to explore theextent to which the JSJ-decomposition of a group can be foundalgorithmically. The specific class of groups to be examined consistsof graphs of free groups, which are of current interest on a number offronts. With Guo-An Diao, the principal investigator has shown thatthe Grushko decomposition (a coarser decomposition than the JSJ) ofthese groups can be found algorithmically. There are two possiblesatisfactory resolutions, namely a construction of an algorithm toproduce the JSJ-decomposition of a group in the class or a proof thatthis problem is unsolvable. For this class of groups, which exists onthe boundary of what can be understood algorithmically, eitherresolution would be important.Geometric group theory is a relatively young branch of mathematicsthat uses geometric methods to understand groups. The idea to take aproblem in group theory, translate it into a problem in geometry, andthen use geometric methods to solve the translated problem. Theprincipal investigator proposes to use geometric methods to exploretwo general areas. The first is motivated by Zlil Sela's geometricsolution to the Tarski problem which on the face of it is a problem inthe intersection of group theory and logic. Roughly, the problem is todistinguish groups using the truth of statements formed using only theusual symbols of logic ("there exists", "for all", variables, etc) andgroup operations. For example, you can tell if a group is abelian byasking whether it is true that, for all elements x and y, xy=yx. Thefirst proposed project, joint with Mladen Bestvina, is to simplify andextend Sela's ideas. This project is ongoing and there has alreadybeen significant progress. The second proposed project follows anothertrend in group theory that is encapsulated by the question: How muchgroup theory can a computer understand? Much as molecules can bedecomposed into simpler pieces (atoms), groups can be often bedecomposed into simpler groups. The project is to understand suchdecompositions for a class of groups called "graphs of free groups", aclass which has attracted much recent attention. It turns out thatdecomposing groups translates into the geometric process of cutting aspace into simpler pieces. The goal is to use this geometric idea todiscover an algorithm that, given a group, finds its decomposition.The principal investigator, with Guo-An Diao, has already constructedan algorithm for a related process that can readily be implemented ona computer.