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**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.)

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.)

Reviewer: Frank Wagner (Villeurbanne)

### MSC:

03C45 | Classification theory, stability, and related concepts in model theory |

03C60 | Model-theoretic algebra |

### Keywords:

superrosy group; interpretable field; dependent theory; NIP; group action on a set of rank 1; non-independence property; hereditarily finitely satisfiable generics; finite thorn rank
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DOI

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