## Abstract

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∈ε}p^{-1} sin^{2}(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 ε_{1} ∼ ε_{2} if and only if G(ε_{1}) = G(ε_{2}), where ε_{1} and ε_{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.

Original language | English (US) |
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Pages (from-to) | 823-843 |

Number of pages | 21 |

Journal | Houston Journal of Mathematics |

Volume | 27 |

Issue number | 4 |

State | Published - 2001 |

## Keywords

- Prime numbers
- Representable groups
- Type III factors

## ASJC Scopus subject areas

- General Mathematics