Research in Geometry and Topology

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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.