Asymptotic and Algorithmic Invariants of Groups

Vanderbilt University, Nashville TN

Investigators

Abstract

The investigators study asymptotic and algorithmic properties of groups. The area of research includes Dehn functions and relative Dehn functions of groups, the complexity of algorithmic problems of groups, Burnside problems for groups. Many papers published during the last forty years showed deep connection between asymptotic invariants of groups and topology, geometry, amenability, dynamical systems, etc. Quite recently, the proposers (jointly with Rips and Birget) discovered a close relationship between the asymptotics of isoperimetric functions, on the one hand, and the complexity of the word problem for groups, on the other hand. The investigators are developing their method, combining geometric and computational approaches. One of their goals is to obtain the solution of other algorithmic problems using geometric group theory. In particular the investigators believe they will be able to prove that every recursively presented finitely generated group can be embedded into a finitely presented group with the same degree of unsolvability of the conjugacy problem. This would solve a problem formulated by Collins in 1976. The computational-geometric methods of the investigators also show promise of being able to be fruitfully applied to the Burnside-type problems. The investigators are close to constructing non-amenable finitely presented groups having no non-cyclic free subgroups. This will solve the corresponding well-known problem which goes back to von Neumann's 1929 paper. Since the group in question is a finitely presented extension of a torsion group of finite exponent by a cyclic group, this example will be a breakthrough in the search for a finitely presented torsion group (one of the main open problem in the theory of infinite groups). It is a well known point of view after Klein, Hilbert, Einstein and Weil that fundamental laws describe symmetries occurring in the nature. The symmetry of an object can be measured by the group corresponding to the object. Groups can be defined 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 symmetries, and describe certain relations between the basic symmetries such that all other relations follow from the chosen relations. One of the main problems about a group given by generators and relations and relations is the word problem: when are two compositions of generators the same? In some exotic cases this problem can be undecidable, that is there are groups for which there are no automatic procedures to recognize if two words of generators are equal. but even in cases when this problem is decidable, the automatic procedure can be very complicated. In recent years the investigators discovered a deep relationship between the word problem of a group and the global geometry of the group. The geometry of a group 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.