The proposal describes several projects related to algorithmic and asymptotic aspects of group theory: * Constructing finitely presented groups with "transcendental" properties (in particular, finitely presented infinite torsion groups). * Complexity of residually finite groups. * Asymptotic and Burnside properties of group actions of maximal growth * Asymptotic cones of finitely presented groups. * Asymptotic properties of amenable groups. These topics are intimately related. For example, Higman embeddings and S-machines are going to be used to construct finitely presented torsion groups and in the study of complexity of the word problem in groups.
Modern group theory is a very fast developing area of mathematics that accumulates methods from geometry, combinatorics, logic and computer science. Our proposal describes projects that are require methods and ideas from all these areas. In particular, constructing a finitely presented infinite torsion group (which is one of the outstanding problems in group theory) will require constructing a suitable kind of Turing machine (called S-machines) and adapting methods from geometric group theory to find suitable quotients of groups simulating these machines. Another idea coming from geometry (and due mostly to Gromov) is the idea of an asymptotic cone of a group. We are going to study asymptotic cones to discover algebraic and algorithmic properties of groups.
