Connected components of definable groups and o-minimality I

We give examples of definable groups G (in a saturated model, sometimes o-minimal) such that G(00) not equal 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 amenable, giving a positive answer to a conjecture of Newelski and Petrykowski in this special case of groups definable in o-minimal structures. (C) 2012 Elsevier Inc. All rights reserved.