Graphs, Trees and Geometric Group Theory

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This project studies automorphism groups and deformation spaces of related metric objects. The motivating examples include the group GL(n,Z) acting on the deformation space of flat n-tori and the group Out(Fn) acting on the deformation space of compact metric graphs with fundamental group Fn. The project has four main components, which take off from these basic examples in various directions. The first component, joint with Martin Bridson, studies rigidity properties of Out(Fn) which limit the possibilities for maps between Out(Fn) and Out(Fm) and constrain the possible actions of Out(Fn) on spheres, contractible manifolds and CAT(0) spaces. The second component, joint with Ruth Charney, targets automorphism groups of right-angled Artin groups. In recent work Charney and the PI have shown that many properties shared by Out(Fn) and GL(n,Z) are in fact shared by the outer automorphism groups of all right-angled Artin groups. The tools used have been largely algebraic, but this project will develop new geometric tools using CAT(0) geometry. The third component, joint with John Smillie, considers deformation spaces of flat surfaces of arbitrary genus. The proposal is to define a bordification of the space of marked translation surfaces with a fixed number of singular points which descends to a compactification of the moduli space of such translation surfaces. This is motivated by analogous bordifications of spaces for SL(n, Z) and Out(Fn) and combines ideas from both, and should be useful in studying cohomological properties of these groups. The fourth part of the project, joint with Jim Conant and Martin Kassabov, returns to Out(Fn) and investigates the rational cohomology of Out(Fn) via the connection found by Kontsevich between this cohomology and the cohomology of a certain Lie algebra associated to the Lie operad.A powerful tool in mathematics is to encode the structure of a geometric object in an algebraic form. One can then use algebraic ideas to study the geometric object, or geometric ideas to study the algebraic object. This proposal uses a bootstrap of this idea to the next level: one can relate the group of {\it transformations}, or {\it automorphisms} of an algebraic object to the space of {\it deformations} of an associated geometric object. The underlying algebraic objects are quite simple: they are free groups, free abelian groups and right-angled Artin groups, but their automorphism groups are remarkably complex and still poorly understood. Similarly, the associated algebraic objects (trees, Euclidean spaces and CAT(0) spaces) are uncomplicated but their deformation spaces exhibit complex behavior, which has implications in many areas of pure and applied mathematics. The project will employ topological and geometric tools to understand these deformation spaces and translate the information obtained into new information about automorphism groups.