Permutation Groups in o-Minimal Structures

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In this paper we develop a structure theory for transitive permutation groups definable in o-minimal structures. We fix an o-minimal structure M, a group G definable in M, and a set Ω and a faithful transitive action of G on Ω definable in M, and talk of the permutation group (G, Ω). Often, we are concerned with definably primitive permutation groups (G, Ω); this means that there is no proper non-trivial definable G-invariant equivalence relation on Ω, so definable primitivity is equivalent to a point stabiliser Gα being a maximal definable subgroup of G. Of course, since any group definable in an o-minimal structure has the descending chain condition on definable subgroups [23] we expect many questions on definable transitive permutation groups to reduce to questions on definably primitive ones. Recall that a group G definable in an o-minimal structure is said to be connected if there is no proper definable subgroup of finite index. In some places, if G is a group definable in M we must distinguish between definability in the full ambient structure M and G-definability, which means definability in the pure group G:= (G, .); for example, G is G-definably connected means that G does not contain proper subgroups of finite index which are definable in the group structure. By definable, we always mean definability in M. In some situations, when there is a field R definable in M, we say a set is R-semialgebraic, meaning that it is definable in (R, +, .). We call a permutation group (G, Ω) R-semialgebraic if G, Ω and the action of G on Ω can all be defined in the pure field structure of a real closed field R. If R is clear from the context, we also just write ‘semialgebraic’.