(joint work with E. Hrushovski and A. Pillay)
If G is a definably compact, connected group definable in an o-minimal structure then, as is known, G/Z(G) is semisimple (no infinite normal abelian subgroup).
We show, that in every o-minimal expansion of an ordered group:
If G is a definably connected central extension of a semisimple group then it is bi-intepretable, over parameters, with the two-sorted structure (G/Z(G), Z(G)). Many corollaries follow for definably connected, definably compact G.
Here are two:
1. (G,.) is elementarily equivalent to a compact, connected real Lie group of the same dimension.
2. G can be written as an almost direct product of Z(G) and [G,G], and this last group is definable as well (note that in general [G,G] is a countable union of definable sets, thus not necessarily definable).