## Fields interpretable in superrosy groups with NIP (the non-solvable case).(English)Zbl 1195.03039

In continuation of [“Fields interpretable in rosy theories”, Isr. J. Math. (to appear)], where the author generalized Cherlin’s analysis of groups of Morley rank 2 to dependent (NIP) groups of thorn-rank 2 with hereditarily finitely satisfiable generics (hfsg), in the paper under review he generalizes part of Hrushovski’s characterisation of group actions on a strongly minimal set from the finite Morley rank to the dependent, finite thorn rank and hfsg context, namely the interpretability of a field. Examples of such groups include superstable groups of finite Lascar rank and definably compact groups definable in an o-minimal expansion of a real-closed field.
Recall that a theory is dependent if in no model there is a formula $$\varphi(\bar x,\bar y)$$ and tuples $$(\bar a_i:i<\omega)$$ and $$(\bar b_I:I\subseteq\omega)$$ such that $$\varphi(\bar a_i,\bar b_I)$$ holds iff $$i\in I$$. It has thorn-rank $$n$$ if a longest strongly dividing chain (in a single variable) has length $$n+1$$. Finally, a group has finitely satisfiable generics (fsg) if there is a global type $$p$$ over the monster model $$G$$ and a small subset $$A\subseteq G$$ such that the translate $$g\cdot p$$ is finitely satisfiable in $$A$$ for all $$g\in G$$; the group has hereditarily finitely satisfiable generics if every definable subgroup is fsg.
Theorem. Let $$G$$ be a definable hfsg group of ordinal thorn-rank in a dependent theory which acts faithfully and definably on a definable set $$S$$ of thorn-rank 1. If $$G$$ is not soluble-by-finite, then there is an infinite field interpretable; if the theory is rosy, then $$G$$ has finite thorn-rank.
The proof is by induction on the thorn-rank in a more general statement, where non-solubility is replaced by the condition that no definable subgroup of the stabilizer of a certain point of $$S$$ has a normalizer of finite index in $$G$$. The base case is then the earlier result on thorn-rank 2 groups; under the hypotheses it cannot be virtually nilpotent and hence equals $$K^+\rtimes K^\times$$ for a definable field $$K$$. (Note that a non-soluble connected group of finite Morley rank acting on a strongly minimal set is $$\text{PSL}_3(K)$$, and hence of rank 3. A similar characterisation remains open in the present context.)

### MSC:

 03C45 Classification theory, stability, and related concepts in model theory 03C60 Model-theoretic algebra
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### References:

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