| Résume | Groups under composition of regular growth rates, together with an ordering or an exponentiation in the sense of Miasnikov-Remeslennikov, naturally appear in o-minimal geometry and asymptotic differential algebra. Yet little is known about their first-order properties. There is no compositional analog of the now well-studied first-order theory of H-fields, and no good theory of extensions of such expansions of groups.
Given a word w(y) over a group G with a single variable y, the existence of a solution to w(y)=1 in an extension of G is in general a difficult problem. It fails even for certain specific types of equations if one wants to preserves certain first-order properties of G, such as orderability. I expect that this question is more traceable within an elementary class of ordered groups that contains certain groups of o-minimal germs. I will explain how to use of valuations on groups, ordered groups and exponential groups as tools to study equations over such groups, and show how one can recover more general results about unary equations over torsion-free groups.
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