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.
A snowflake can be put into 12 different positions which look identical. The same is true of a tetrahedredron, but its symmetries are fundamentally different in a way which is not caught by merely counting them. For example, there is a rotation which can be repeated six times before a snowflake returns to its original position, but there is no transformation of a tetrahedron which requires six iterations before the tetrahedron comes back to where it started. The proper measure of symmetry is a mathematical object called a group, in the same way that the proper measure of quantity is a number. Symmetry groups play a central role in all of mathematics. In addition to its aesthetic appeal, symmetry can simplify computations, rule out pathologies and generally bestow a deeper understanding of any mathematical or physical system. Groups themselves have symmetries, called automorphisms. A group G is already an abstract mathematical object, and thinking about the group of Aut(G) of automorphisms of G adds another level of abstraction. The main objective of my research is to understand these automorphism groups, and the main tools come from geometry. The idea is that given an automorphism group Aut(G) one looks for a concrete geometric object whose symmetries can be identified with Aut(G); this enables one to visualize Aut(G) and study it using geometric tools. This might be compared to exhibiting seven apples to explain the abstract concept of "seven." Manipulating the apples can then help one understand attributes of the number seven, such as which numbers are less than seven, whether seven is evenly divisibly by two, etc. Paradoxically, the simplest groups often have the most complex and interesting automorphism groups. One of the simplest of all (infinite) groups is called a free group; it is free because no generator has any relation with any other generator. In early work with Marc Culler I built a geometric model for the automorphism group of a free group F with n generators. Technically it’s really a model for the group Out(F) of outer automorphisms of F, where the outer automorphism group encodes the interesting part of the automorphism group. This explains the fact that this geometric model is now called Outer space. By now Outer space has been used by a great many people in addition to Culler and myself to solve problems about the structure of Out(F). The first major objective of this project was to replicate this success for a wider class of groups than free groups, called right-angled Artin groups or RAAGs for short. In free groups generators never commute. In RAAGs some generators are allowed to commute but there are no other relations among them. Thus RAAGs are slightly more complicated than free groups but not much, and they still have large and interesting automorphism groups. Ruth Charney and I succeeded in constructing an "outer space" for RAAGs analogous to the Outer space for free groups mentioned above. This opens the door to studying general automorphism groups of RAAGs using geometric methods which have been successful for studying Out(F). A second major outcome of the project was to uncover surprising new connections between Out(F) (and Outer space) and other areas of mathematics, specifically number theory and the theory of continuous (i.e. Lie) groups. This was accomplished by studying algebraic invariants called cohomology rings. Establishing such connections between fields is not only exciting, but injects new ideas into each field and advances our understanding in both directions.
