  We describe the Zariski-closure of sets of torsion points in connected
algebraic groups. This is a generalization of the Manin-Mumford conjecture for
commutative algebraic groups proved by Hindry. He proved that every subset with
Zariski-dense torsion points is the finite union of torsion-translates of
algebraic subgroups. We formulate and prove an analogous theorem for arbitrary
connected algebraic groups.
  We also define a canonical height on connected algebraic groups that
coincides with a N\'eron-Tate height if $G$ is a (semi-) abelian variety. This
motivates a generalization of the Bogomolov conjecture to arbitrary connected
algebraic groups defined over a number field. We prove such a generalization as
well.
