Asymptotic invariants of groups

Vanderbilt University, Nashville TN

Investigators

Abstract

The investigators study asymptotic invariants of groups. The topics include: constructing finitely presented groups with "transcendental" properties (infinite torsion and with Kazhdan property (T) ), residual properties and linearity of hyperbolic groups, groups with irregular or small Dehn functions and other filling functions of groups, asymptotic cones of finitely presented groups, and asymptotic invariants of embeddings into Hilbert spaces (Hilbert space compressions of groups). These topics are intimately related. For example, Higman embeddings and S-machines are used to construct finitely presented torsion groups, in the study of Dehn functions, and in the study of residual properties of hyperbolic groups. The investigators propose to organize several group theory conferences and attract students to their area of mathematics. They are also going to give several courses on the topic of the proposal. It is a well known point of view after Klein, Hilbert, Einstein and Weil that fundamental laws describe symmetries occuring in the nature. The symmetry of an object can be measured by the group corresponding to the object. Groups can be difined either as groups of symmetries or abstractly by an algorithmic description (generators and relations). In the second approach, the investigators choose some basic symmetries (generators) so that all other symmetries are compositions (words) of the basic ones, and describe certain relations between the basic symmetries such that all other relations follow from the chosen ones. The geometry of groups given by such presentations is described in terms of certain asymptotic invariants. The invariants have been known since the pioneering works of M.Dehn at the beginning of the 20th century, but the investigators discovered deep relationship between these invariants and algorithmic problems. The investigators are developing their geometric method solving old mathematical problems of algorithmic nature and corresponding algebraic problems about groups.