ON A CLASS OF SUBGROUPS OF R ASSOCIATED WITH SUBSETS OF PRIME NUMBERS

Abstract. A subgroup G of (R,+) is called representable if there exists a set of prime numbers E such that G = G(E) = ¨t ∈ R;∑p∈E p−1 sin2(t log p) <∞©, 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 sub-groups 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 ⊂ R and any countable subset Σ of the complement of H in R there exists a representable group Γ which contains H and does not intersect Σ. 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 E1 ∼ E2 if and only if G(E1) = G(E2), where E1 and E2 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 de-rived as a corollary of the vanishing of the first cohomology group on certain open sets