  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 always has maximal
definably compact subgroups, and for every such a K there is a maximal
definable torsion-free subgroup H such that G/N can be decomposed as G/N = KH,
and the intersection between K and H is trivial. Thus G is definably homotopy
equivalent to K. When G is solvable then G/N is already definably compact. In
any case (even when G has no maximal definably compact subgroup) we find a
definable Lie-like decomposition of G where the role of maximal tori is played
by maximal 0-subgroups.

Logic