  We show that the Witten genus of a string manifold $M$ vanishes, if there is
an effective action of a torus $T$ on $M$ such that $\dim T>b_2(M)$. We apply
this result to study group actions on $M\times G/T$, where $G$ is a compact
connected Lie group and $T$ a maximal torus of $G$.
  Moreover, we use the methods which are needed to prove these results to the
study of torus manifolds. We show that up to diffeomorphism there are only
finitely many quasitoric manifolds $M$ with the same cohomology ring as
$#_{i=1}^k \pm\mathbb{C} P^n$ with $k<n$.
