Asymptotic and Algorithmic Invariants of Groups

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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. Quiterecently, the proposers (jointly with Rips and Birget) discovered a closerelationship between the asymptotics of isoperimetric functions, on the onehand, and the complexity of the word problem for groups, on the other hand.The investigators are developing their method, combining geometric andcomputational approaches. One of their goals is to obtain the solutionof other algorithmic problems using geometric group theory. In particularthe investigators believe they will be able to prove that every recursively presentedfinitely generated group can be embedded into a finitely presented groupwith the same degree of unsolvability of the conjugacy problem. This wouldsolve 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 presentedgroups having no non-cyclic free subgroups. This will solve thecorresponding well-known problem which goes back to von Neumann's 1929paper. Since the group in question is a finitely presented extension of atorsion group of finite exponent by a cyclic group, this example will bea breakthrough in the search for a finitely presented torsiongroup (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 Weilthat fundamental laws describe symmetries occurring in the nature. Thesymmetry of an object can be measured by the group corresponding to theobject. Groups can be defined as groups of symmetries or abstractly by analgorithmic description (generators and relations). In the second approach,the investigators choose some basic symmetries (generators) so that all other symmetriesare compositions (words) of the basic symmetries, and describe certainrelations between the basic symmetries such that all other relations followfrom the chosen relations. One of the main problems about a group given bygenerators and relations and relations is the word problem: when are twocompositions of generators the same? In some exotic cases this problem canbe undecidable, that is there are groups for which there are no automaticprocedures to recognize if two words of generators are equal. but evenin cases when this problem is decidable, the automatic procedure can be verycomplicated. In recent years the investigators discovered a deeprelationship between the word problem of a group and the global geometry ofthe group. The geometry of a group is described in terms of certainasymptotic invariants. The invariants have been known since the pioneeringworks of M. Dehn at the beginning of the 20th century but the investigatorsdiscovered deep relationship between these invariants and algorithmicproblems. The investigators are developing their geometric method solvingold mathematical problems of algorithmic nature and corresponding algebraic problems about groups.