So these lecture notes contain this statement (Exercise 3.3.5):
If $T$ is a torus acting on a compact manifold $M$ such that every isotropy subgroup has codimension greater than one, then there exists a circle inside $T$ that acts freely on $M$.
Could someone point me to a proof? This doesn't seem to be very straightforward. I would imagine this has a lot to do with finiteness of number of orbit types on $M$ (perhaps we don't even need $M$ to be a manifold, if we a priori know that there are only finitely many orbit types?). Also, can we weaken the assumption about codimension of isotropy groups to the action being fixed point free?