On a class of subgroups of R associated with subsets of prime numbers

A subgroup G of (R, +) is called representable if there exists a set of prime numbers E such that G = G( E) = {t is an element of R; Sigma(pis an element ofepsilon) p(-1) sin(2)(t log p) < &INFIN;}, or equivalently if it coincides with Connes' modular T-group associated to a certain ITPFI factor. By our previous work, {0}, R and the cyclic subgroups of R are representable. Moreover, this class of subgroups is closed under homotheties. It also has the important feature of separating countable subgroups of R from countable subsets of their complements, that is for any countable subgroup H &SUB; R and any countable subset &USigma; of the complement of H in R there exists a representable group &UGamma; which contains H and does not intersect &USigma;. In this paper we define and study a natural topology on the space of representable subgroups. This space coincides with the set of equivalence classes of the relation E-1, E-2 if and only if G(E-1) = G( E 2), where E-1 and E-2 are infinite sets of prime numbers. The structure of the homeomorphisms of this space and the cohomology of a certain natural sheaf are being investigated. A stronger version of the separation property is derived as a corollary of the vanishing of the first cohomology group on certain open sets.