Submission date: 4 July 2010.

Abstract:

(Published in: Journal of Symbolic Logic (75), 372-386, 2010.) Let G be a group definable in a monster model \C of a rosy theory satisfying NIP. Assume that G has hereditarily fsg, and 1<\uth(G)<\infty, where \uth(G) is the thorn U-rank of G. We prove that if G acts definably on a definable set of thorn U-rank 1, then, under some general assumption about this action, there is an infinite field interpretable in \C. We conclude that if G is not solvable-by-finite and it acts faithfully and definably on a definable set of thorn U-rank 1, then there is an infinite field interpretable in \C. As an immediate consequence, we get that if G has a definable subgroup H such that \uth(G)=\uth(H)+1 and G/\bigcap_{g \in G}H^g is not solvable-by-finite, then an infinite field interpretable in \C also exists.

Mathematics Subject Classification: 03C45, 03C60

Keywords and phrases: superrosy group, interpretable field, non independence property

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