Interaction of Algebraic, Algorithmic and Asymptotic Properties in Finitely Generated Groups

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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 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 broader impact of the project will affect students at all levels, and professional investigators. The principal investigators will continue running Nashville Math Club for K-12 students, engaging undergraduate students in research projects, and advising graduate students. In this research the principal investigators will work on several fundamental problems together with ideas of their solutions related to Burnside, asymptotic and algorithmic problems of finitely generated groups. The problems are intrinsically connected by the proposed methods of solutions and by the tools used in the proposed approaches in their solutions. The project for a definition of a proper analogue of asymptotic cones for Golod-Shafarevich groups has a potential impact far beyond group theory. Other problems concern the existence of an infinite finitely presented torsion group for which a detailed approach for a positive solution is provided, the existence of amenable groups with non-trivial Morse boundary, and the construction of a finitely presented group with maximal possible (continuum) different asymptotic cones (assuming the Continuum Hypothesis).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.