## Sous-groupes minimaux des groupes de Lie commutatifs réels, et applications arithmétiques. (Minimal subgroups of commutative real Lie groups and arithmetic applications).(French)Zbl 0672.10024

Let R be a commutative real Lie group. We say that a subgroup $$\Gamma$$ of R is a minimal subgroup of R if it is dense in R, finitely generated, and if no subgroup of $$\Gamma$$ of rank strictly less than the rank of $$\Gamma$$ is dense in R. In this paper, we show that the rank of such a subgroup is bounded by a constant which depends only on R. In the case where R is $$({\mathbb{R}}^{\times})^ p\times ({\mathbb{C}}^{\times})^ q,$$ this bound is $$2p+3q.$$ The main result is that if $$\Gamma$$ is a minimal subgroup of $$({\mathbb{R}}^{\times})^ p\times ({\mathbb{C}}^{\times})^ q$$ contained in $$({\bar {\mathbb{Q}}}^{\times}\cap {\mathbb{R}}^{\times})^ p\times ({\bar {\mathbb{Q}}}^{\times})^ q$$ then its rank is $$p+q+1$$ for $$q=0$$, $$1\leq p\leq 4$$ and for $$q=1$$, $$p\leq 1$$. This gives a partial answer to a question of J. J. Sansuc.
Reviewer: D.Roy

### MSC:

 11J72 Irrationality; linear independence over a field 11J81 Transcendence (general theory) 22E15 General properties and structure of real Lie groups

### Keywords:

commutative real Lie group; minimal subgroup; rank
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