A New Metric Criterion for Non-amenability I

By studying the so-called traveling salesman groups, we obtain a new metric criterion for non-amenability. As an application, we give a new and very short proof of non-amenability of free Burnside groups with sufficiently big odd exponent. Amenable groups have been introduced by John von Neumann in 1929 in connection with Banach-Tarski Paradox, although earlier, Banach himself had understood that, for example, the group Z is amenable. J.von Neumann’s original definition states that a discrete group is amenable iff it admits an additive invariant probability measure. In 1950’s Folner gave a criterion which, for finitely generated groups, can be expressed in terms of the Cayley metric of the group. Using this criterion it is very easy to see that abelian groups are amenable and non-abelian free groups (e.g. F2) are not amenable. In 1980’s Grigorchuk [Gr1] introduced a new metric criterion for amenability in terms of the co-growth as a refined version of Kesten’s criterion [K]. Using this criterion, A.Ol’shanskii [Ol1] constructed a counterexample to von Neumann Conjecture, and using the same criterion S.Adian [Ad1] proved that free Burnside groups of sufficiently large odd exponent are non-amenable. Recently, A.Ol’shanskii and M.Sapir [OS] found a finitely presented counterexample to von Neumann Conjecture where they establish non-amenability using the same criterion again. Based on our studies of traveling salesman groups, we introduce a different metric criterion (sufficient condition) for non-amenability. Roughly speaking, this criterion is based on verifying that certain type of words in a group do not have small length. This, in a lot of cases, seems manageable to show, compared to measuring the boundary of a finite set. More than that, it was proved(observed) by von Neumann that if a group contains a copy of F2 then it is non-amenable. But if the group contains no subgroup isomorphic to F2 then it is usually hard to prove non-amenability. However, one can continue in the spirit of von Neumann’s observaton. For example, from Folner’s criterion it easily follows that if a group is quasi-isometric to a group which contains a copy of F2 then the group is still non-amenable. Our criteria generalizes much further in this direction. Traveling salesman (TS) groups were introduced in [Ak1] and [Ak2]. These groups are very useful and play an important role in the constructions showing that the property of containing free subgroup, containing free subsemigroup, satisfying a law, or a girth type, and numerous other properties are not invariant under quasiisometry. (see [Ak3]). Negatively curved groups turn out to be TS (see below). Here, the term ”negatively curved” is used in a more general sense than being non-elementary word