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We give examples of definable groups G (in a saturated model, sometimes o-minimal) such that G 00 = G 000 , yielding also new examples of " non G-compact " theories. We also prove that for G definable in a (saturated) o-minimal structure, G has a " bounded orbit " (i.e. there is a type of G whose stabilizer has bounded index) if and only if G is definably… (More)

In this sequel to [3] we try to give a comprehensive account of the " connected components " G 00 and G 000 as well as the various quotients G/G 00 , G/G 000 , G 00 /G 000 , for G a group definable in a (saturated) o-minimal expansion of a real closed field. Key themes are the structure of G 00 /G 000 and the problem of " exactness " of the G → G 00… (More)

We study analogues of the notions from Lie theory of Levi subgroup and Levi decomposition, in the case of groups G definable in an o-minimal expansion of a real closed field. With suitable definitions, we prove that G has a unique maximal ind-definable semisimple subgroup S, up to conjugacy, and that G = R · S where R is the solvable radical of G. We also… (More)

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