A REDUCTION TO THE COMPACT CASE FOR GROUPS DEFINABLE IN O-MINIMAL STRUCTURES

Let ${\cal N}\left( G \right)$ be the maximal normal definable torsion-free subgroup of a group G definable in an o-minimal structure M. We prove that the quotient $G/{\cal N}\left( G \right)$ has a maximal definably compact subgroup K, which is definably connected and unique up to conjugation. Moreover, we show that K has a definable torsion-free complement, i.e., there is a definable torsion-free subgroup H such that $G/{\cal N}\left( G \right) = K \cdot H$ and $K\mathop \cap \nolimits^ \,H = \left\{ e \right\}$ . It follows that G is definably homeomorphic to $K \times {M^s}$ (with $s = {\rm{dim}}\,G - {\rm{dim}}\,K$ ), and homotopy equivalent to K. This gives a (definably) topological reduction to the compact case, in analogy with Lie groups.