Award: DMS 1406167, Principal Investigator: Mark E. Feighn
Two projects are proposed in the area of geometric group theory, a relatively young subfield of mathematics where problems from other areas of math are reformulated in geometric terms and then (hopefully) solved using a geometer's toolkit. This approach has been successful in solving problems in such diverse areas as combinatorial group theory (think of a situation such as Rubik's cube where a discrete set of moves is allowed) and logic. Both projects are focused on a particular group of classical interest called "the outer automorphism group of a free group" which is denoted Out(F). Groups arise as sets of symmetries of objects and Out(F) is the set of symmetries of a free group F, an important group from which all others can be constructed. Completion of either project would represent a major advance in the field.
The first project, joint with Mladen Bestvina at the University of Utah, is to continue the study of the geometry of Out(F). In particular, we have a plan to show that Out(F) has finite asymptotic dimension. Part of this work will also be joint with Patrick Reynolds at the University of Utah. The geometry of Out(F) is currently a hot topic with interest spurred in large part by results of Bestvina-Feighn and Handel-Mosher that certain spaces on which Out(F) acts are hyperbolic. The second project, joint with Michael Handel at Lehman College, is to show that Out(F) has a solvable conjugacy problem. The conjugacy problem is a famous decidability question formulated by Max Dehn around 1911 that can be asked about any group. The fact that it remains open for Out(F) reveals a gap in our basic understanding of this group.
