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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 the last twenty years many authors have investigated the analogies between groups definable in o-minimal structures and real Lie groups (see [Ot] for a survey). By a theorem of Pillay ([Pi1]), every definable group G in an o-minimal structure M can be equipped with a topology τ which makes it a topological group. Moreover, if n ∈ N is the o-minimal… (More)
In [KKMZ02] the authors gave a valuation theoretic characterization for a real closed field to be κ-saturated, for a cardinal κ ≥ ℵ 0. In this paper, we generalize the result, giving necessary and sufficient conditions for certain o-minimal expansion of a real closed field to be κ-saturated.
There are strong analogies between groups definable in o-minimal structures and real Lie groups. Nevertheless, unlike the real case, not every definable group has maximal definably compact subgroups. We study definable groups G which are not definably compact showing that they have a unique maximal normal definable torsion-free subgroup N ; the quotient G/N… (More)
In this sequel to  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)
We prove the definability, and actually the finiteness of the commu-tator width, of many commutator subgroups in groups definable in o-minimal structures. It applies in particular to derived series and to lower central series of solvable groups. Along the way, we prove some generalities on groups with the descending chain condition on definable subgroups… (More)