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 G00≠G000, 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.