  We introduce and develop the model-theoretic notions of absolute
connectedness and type-absolute connectedness for groups. We prove that groups
of rational points of split semisimple linear groups (that is, Chevalley
groups) over arbitrary infinite fields are absolutely connected and
characterize connected Lie groups which are type-absolutely connected. We prove
that the class of type-absolutely connected group is exactly the class of
discretely topologized groups with trivial Bohr compactification, that is the
class of minimally almost periodic groups. As an application we generalize some
results of Conversano-Pillay and construct a group $G$ where G^{00}/G^{\infty}
is far from being abelian.
