Kazhdan's property (T), Golod-Shafarevich groups and Kac-Moody groups

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).This project will investigate the structure of several classes of groups defined by generators and relations, including Golod-Shafarevich groups and Kac-Moody groups. The Principal Investigator will study various asymptotic invariants for these groups (e.g., subgroup growth, rank gradient) and certain aspects of their representation theory with the emphasis on Kazhdan's property (T). The methods used in the project involve techniques from different areas, including combinatorial group theory, pro-p groups, arithmetic groups and Lie algebras. The findings of this project will likely have applications beyond group theory, e.g., in the areas of three-manifold topology and graph theory.Groups play a fundamental role in mathematics by describing symmetries of various objects like geometric figures or number systems. A group can often be presented by generators and relations which provide a simple way to define the group but usually offer little insight into its structure. The project will develop new tools that can help better understand a group based on its presentation by generators and relations which, in turn, can yield new information about the object whose symmetries the group describes.