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

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A subgroup G of (R,+) is called representable if there exists a set of prime numbers ε such that G = G(ε) = {t ∈ R; ∑p∈εp-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 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 ⊂ 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 ε1 ∼ ε2 if and only if G(ε1) = G(ε2), where ε1 and ε2 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.

Original languageEnglish (US)
Pages (from-to)823-843
Number of pages21
JournalHouston Journal of Mathematics
Issue number4
StatePublished - 2001


  • Prime numbers
  • Representable groups
  • Type III factors

ASJC Scopus subject areas

  • General Mathematics


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