A fundamental technique in group theory is to study groups by examining their actions on topological spaces. If one restricts attention to spaces with strong geometric properties (such as metric trees or CAT(0) spaces), one can often make the set of actions of G into a topological space itself, called a deformation space, with good topological properties. The automorphism group of G acts on the deformation space by twisting each action to get a new action. In this project we study automorphism groups of various classes of groups by studying their actions on deformation spaces. For free groups, the relevant deformation space, originally defined by the PI and M. Culler, is known as Outer space. A fairly recent development in this subject is that Morita used a connection found by Kontsevich between Outer space and a certain infinite-dimensional symplectic Lie algebra to detect cocycles for Out(F_n), the group of outer automorphisms of the free group of rank n. The PI and J. Conant have reinterpreted these cocycles as cycles on the quotient of Outer space by the action of Out(F_n), and plan to use these to study the unstable cohomology of Out(F_n). A second goal of this project, joint with R. Charney, is to develop an analog of Outer space for right-angled Artin groups, a class which includes both free groups and free abelian groups. A third component is motivated by previous work of the PI on stability of the homology of Out(F_n) and Aut(F_n). Techniques developed for proving these stability results apply to other sequences of groups related to low-dimensional topology, and the PI will further investigate these applications. Other ongoing projects include a study of the space of phylogenetic trees and an investigation of rigidity properties of Out(F_n).
A phenomenon which occurs throughout mathematics and the sciences is that complicated structures can often be understood more easily by codifying the information they contain in terms of graphs and trees. On example of this is the use of graphs and trees to describe the possible splittings of an algebraic object called a group into simpler pieces. When there are many possible splittings, one can construct deformation spaces which measure the ambiguity. In this project we study groups by considering actions of the symmetries of the group on such deformation spaces.
