Executive Summary : | This research proposal focuses on geometric group theory, which studies discrete groups using geometry, topology, and dynamics. It emphasizes the role of negative curvature in understanding a diverse collection of groups, particularly the group of outer automorphisms of a free group, Out(Fn), and groups acting on CAT(0) spaces. Nonpositive curvature is crucial in geometric group theory, whether at large scale like hyperbolic groups or at a small scale like CAT(0) groups. Coxeter groups are an example of groups acting geometrically on CAT(0) spaces, with infinitely many groups even up to Gromov's coarse notion of 'quasi-isometry'. The proposal also explores other interesting classes of groups that don't act as strongly as geometrically on nonpositively curved spaces but admit weaker actions. Two important examples are the mapping class group of a hyperbolic surface and the group of outer automorphisms of free groups. The proposal seeks to expand this dictionary by considering individual free group automorphisms and hyperbolic Out(Fn)-complexes. It also investigates the existence of CAT(0) structures on the free factor complex, one of the most well-understood hyperbolic complexes that Out(Fn) acts on. |