Van Kampen diagrams, machines, and asymptotic behavior of groups
by
Alexander Olshanskii
Vanderbilt and also Moscow State
Coauthors: M.V.Sapir

Mostly I will speak on the results obtained jointly with M.V.Sapir after I moved to the Vanderbilt in 1999.

Consider a finite presentation of a group G in terms of generators and relators: < A, R > . Then for every word w in the group alphabet A, vanishing in G, there is a planar van Kampen diagram responsible for some deduction of the equality w=1 from the defining relations r=1 where r ∈ R. The Dehn function d(n) gives the upper bound of areas of (minimal) diagrams whose perimeters ≤ n. Up to a natural equivalence, it does not depend on the choice of the finite presentation for G. The asymptotic behavior of f(n) is an important invariant of a finitely presented group G, connected to geometric and algorithmic properties of G.

It is not difficult to prove that every Dehn function is a time function of a Turing machine, but, unfortunately, the converse claim is false, and to investigate Dehn functions one is to work hard and discover new properties of diagrams and new types of them. Our method presents answers to a number of long-standing problems in Group Theory, in particular, on the complexity of computations in groups, on the algorithmic conjugacy problem, and on a problem about amenability of finitely presented groups. Our recent papers present examples of groups with strange behavior of their Dehn functions. These results are also applicable to the theory of asymptotic cones of groups.

Date received: March 19, 2006