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Let $K$ be an algebraically closed valued field which is $C$-minimal, as defined, for example, in this article. Examples include pure algebraically closed valued fields, as well as Lipschitz and Robinson's expansions by rigid subanalytic sets (Lipshitz. Rigid subanalytic sets. Amer. J. Math., 115(1):77-108, 1993)

Is it true then that the structure induced on the value group of $K$ is that of a vector space (over a field that depends on $K$)? This would mean, in particular, that definable sets are Boolean combinations of sets defined by systems of affine inequalities (i.e. polyhedra).

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I think the answer is no. Consider an algebraically closed valued field in the three sorted language. Using a relative quantifier elimination argument, any o-minimal expansion of the value group preserves $C$-minimality of the structure (meaning, definable subsets in one variable of the field sort are finite boolean combinations of balls). For instance, if your value group is the reals, putting the exponential function in the value group sort provides a counterexample.

Let me mention that a result of Cluckers in this article shows that your question is almost true for $P$-minimal expansions of $p$-adically closed fields where the 'almost' means that you need to allow congruence relations: the induced structure on the value group is precisely the Presburger structure.

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  • $\begingroup$ Hi Pablo! Thank you for the counterexample. I wonder if one can formulate a condition similar to polynomial boundedness for C-minimal fields that would guaruantee the vector space structure on the value group (I am drawing parallel to results of van den Dries and Lowenberg for tame expansions of polynomially bounded o-minimal structures here) $\endgroup$
    – Dima Sustretov
    Commented Jul 17, 2016 at 15:13
  • $\begingroup$ I would be rather happy to find such a condition. You can try polynomially boundedness, but unfortunately there is no (as far as I know) dichotomy as in o-minimality where either you are polynomially bounded or the exponential function is definable in your structure (this is a theorem of Miller). $\endgroup$
    – Cubikova
    Commented Jul 19, 2016 at 6:40

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