Research in geometric group theory

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Three rather ambitious projects are proposed in the area of geometricgroup theory. The first is inspired by Zlil Sela's geometric solutionof Tarski's logic problem. The PI and Mladen Bestvina plan to completetheir program to solve by geometric means two forty-year-old problemsof Mal'cev. They also propose to generalize their methods and providean alternate proof to the Tarski problem. The second project, jointwith Michael Handel, is to solve the conjugacy problem for the outerautomorphism group Out(F) of a free group F. Much of the research inthis area is driven by the similarities between linear groups, surfacemapping class groups, and Out(F). The conjugacy problem has beensolved for these other two classes. The final project, again jointwith Mladen Bestvina, is to better understand the geometry of outerautomorphism groups of free groups. In particular, it is proposed to showthat a complex analogous to the curve complex in the mapping classgroup setting is word hyperbolic.Three projects are proposed in the area of geometric grouptheory. Geometric group theory is a relatively young branch ofmathematics in which problems from other areas of mathematics arereformulated and then solved in geometric terms. This approach hasbeen successful in areas that are sometimes viewed as distant fromgeometry. For example, Zlil Sela used geometric methods to solve anold logic problem of Tarski. Inspired by Sela's methods, the PI andMladen Bestvina propose to solve two old logic problems ofMal'cev. Group theory is another area where these methods have beenparticularly successful. Groups are ubiquitous in math and thesciences. The set of symmetries of a molecule is an example. There isan important class of groups called "free groups" from which all othergroups can be constructed. The set of symmetries of a free group F isanother important group, denoted Out(F), which has been the subject ofmuch current interest. The PI proposes two other projects focusing onOut(F). In one, he and Michael Handel propose to solve an old andfundamental problem on Out(F), namely that its conjugacy problem issolvable. In the other project, the PI and Bestvina propose to betterunderstand the intrinsic geometry of Out(F).