  In this article we study Semi-abelian analogues of Schanuel conjecture. As
showed by the first author, Schanuel Conjecture is equivalent to the
Generalized Period Conjecture applied to 1-motives without abelian part.
Extending her methods, the second, the third and the fourth authors have
introduced the Abelian analogue of Schanuel Conjecture as the Generalized
Period Conjecture applied to 1-motives without toric part. As a first result of
this paper, we define the Semi-abelian analogue of Schanuel Conjecture as the
Generalized Period Conjecture applied to 1-motives. C. Cheng et al. proved that
Schanuel conjecture implies the algebraic independence of the values of the
iterated exponential and the values of the iterated logarithm, answering a
question of M. Waldschmidt. The second, the third and the fourth authors have
investigated a similar question in the setup of abelian varieties: the Weak
Abelian Schanuel conjecture implies the algebraic independence of the values of
the iterated abelian exponential and the values of an iterated generalized
abelian logarithm. The main result of this paper is that a Relative
Semi-abelian conjecture implies the algebraic independence of the values of the
iterated semi-abelian exponential and the values of an iterated generalized
semi-abelian logarithm.
