In spite of a significant progress on the topology of the automorphism group of a free group, relatively little is known about the large scale geometry of this group. A part of the proposal is a program to remedy the discrepancy with mapping class groups, where the geometry is very well understood, thanks to the recent work of Masur, Minsky and others. The first step in this program is hyperbolicity of the free factor complex. Other questions proposed are finite presentability of Torelli groups, finiteness properties of arithmetic groups over function fields and quasi-isometric rigidity of right-angled Artin groups.
The interplay between algebra and geometry is one of the classical themes in mathematics. For example, symmetries of Platonic solids form fundamental examples of groups. In geometric group theory the situation is reversed: one starts with a group of symmetries of an algebraic object (for example, another group) and constructs a geometric object with the same symmetries. The study of the geometry of this object then leads to a greater understanding of the algebraic object one started with. The proposal contains several instances of this approach. Culler-Vogtmann's Outer space is a geometric object whose symmetries are the same as those of a free group (of finite rank). Recent results reveal that this space has negative curvature in some directions, much like the surface of a saddle. Understanding this phenomenon better would lead to a greater understanding of the symmetry group of the free group.
The work in this proposal is about understanding the symmetry groups. The bulk is about the particular symmetry group, namely of the free group. The goal is to understand the geometry of this algebraic object. The geometry that is particularly appealing and well understood is hyperbolic geometry. The main outcome of this research is that hyperbolic geometry plays a critical role in the understanding of the group of symmetries of the free group. In addition, certain additional hyperbolic aspects were found, beyond what was already known, of the so called mapping class group -- this is the group of symmetries of surfaces. In addition, the research involved understanding what the associated hyperbolic space looks like at infinity. Remarkably, it turned out that it is very closely related to the space of graphs studied for a long time, the so called Outer space. Finally, the research included understanding of approximate actions of a group on certian kind of infinite dimensional vector spaces. It turns out that these are much easier to understand than the actual actions, and provide a lot of information about the given group. The PI advised 3 PhD students who finished in the reporting period, and two of these were female. In addition, the PI advised a number of postdocs.
