On The Ore Condition For The Group Ring Of R. Thompson'S Group F

Let R=K[G] be a group ring of a group G over a field K. The Ore condition says that for any a,b is an element of R there exist u,v is an element of R such that au = bv, where u 0 or v not equal 0. It always holds whenever G is amenable. Recently it was shown that for R. Thompson's group F the converse is also true. So the famous amenability problem for F is equivalent to the question on the Ore condition for the group ring of the same group.It is easy to see that the problem on the Ore condition for K[F] is equivalent to the same property for the monoid ring K[M], where M is the monoid of positive elements of F. In this paper we reduce the problem to the case when a, b are homogeneous elements of the same degree in the monoid ring. We study the case of degree 1 and find solutions of the Ore equation. For the case of degree 2, we study the case of linear combinations of monomials from S={x(0)(2),x(0)x(1),x(0)x(2),x(12),x(1)x(2)}. This set is not doubling, that is, there are nonempty finite subsets X subset of M subset of F such that |SX|<2|X|. As a consequence, the Ore condition holds for linear combinations of these monomials. We give an estimate for the degree of u, v in the above equation.The case of monomials of higher degree is open as well as the case of degree 2 for monomials on x(0),x(1), ...,xm, where m >= 3. Recall that negative answer to any of these questions will immediately imply non-amenability of F.