9626699 Feighn In the current project, the Principal Investigator will continue his work, joint with M. Bestvina and M. Handel, on the structure of the outer automorphism group of a free group. The methods used are a mixture of techniques from E. Rips's theory of real trees and from the train track theory of Bestvina and Handel. The Principal Investigator will also examine to what extent the JSJ theorem of Rips and Z. Sela can be extended to splittings over other than cyclic groups. Now is an exciting time to be a geometric group theorist. Much beautiful mathematics is devoted to classification results. For example, the classification of surfaces (spaces, such as the sphere, that are locally the same as a plane) is classical. This classification is akin to Mendeleev's periodic table of the elements -- each surface appears once and only once. In recent years, due in large part to the geometric ideas of W. Thurston, there has been great progress towards a classification theorem for 3-manifolds (spaces that are locally like 3- dimensional space). Inspired by these manifold successes, mathematicians have begun to look at groups from a geometric viewpoint. (A group is a fundamental object found throughout mathematics and the sciences. Roughly speaking, a group is the set of symmetries of some object.) Mathematics has arrived at the point where the possibility of a classification theorem for (geometric) groups can be envisioned. For example, a recent result of E. Rips and Z. Sela describes how to break a group into more fundamental pieces. In the current project, the Principal Investigator will continue his research along these lines. More specifically, he will show how to break up groups into fundamental pieces in ways other than that of Rips and Sela. Also, he will continue his work with M. Bestvina and M. Handel undertaking a detailed study of some of the groups that will play a key role in this classification. ***
