  We prove several structural results on definably compact groups G in
o-minimal expansions of real closed fields, such as (i) G is definably an
almost direct product of a semisimple group and a commutative group, and (ii)
the group (G, .) is elementarily equivalent to (G/G^00, .). We also prove
results on the internality of finite covers of G in an o-minimal environment,
as well as the full compact domination conjecture. These results depend on key
theorems about the interpretability of central and finite extensions of
definable groups, in the o-minimal context. These methods and others also yield
interpretability results for universal covers of arbitrary definable real Lie
groups, from which we can deduce the semialgebraicity of finite covers of Lie
groups such as SL(2,R).
