Asymptotic Methods in Geometric Group Theory

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Modern group theory is a very fast developing area of mathematics that uses methods from algebra, geometry, combinatorics, topology, probability, logic, and computer science. Applications of group theory are ubiquitous throughout science, from physics and chemistry to cyber security. This award supports collaborative research in the area of asymptotic methods in group theory, an area that has been extremely active since seminal papers by Gromov on hyperbolic groups and asymptotic invariants of groups. This project will significantly advance knowledge of the area by solving several open problems. The investigators will advise graduate students, organize research experiences for undergraduate students, and organize conferences on the subject of asymptotic methods in group theory.The problems under study in this research project include* estimating Dehn functions of metabelian groups;* constructing actions of hyperbolic groups resembling Tarski monster groups;* constructing simple groups that admit highly transitive actions;* describing all possible Tarski numbers of groups; and* finding an amenable group with all asymptotic cones tree-graded.The results of the project will advance understanding of asymptotic methods in group theory.