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?
Suppose that a compact group $G$ acts on a compact manifold $M$, We say that the orbits of $x$ and $y$ have the same type if and only if $G_x$ is conjugated to $G_y$ where $G_x$ is the stabilizer of $x$. There exists a finite number of type since $M$ is compact. Suppose now that $G$ is the torus $T^n$, a type of an orbit is characterized by a subgroup $H_i$ of $T^n$ since $H_i=gH_ig^{-1}$. It the dimension of $H_i< n$, there exists a morphism $f:S^1\rightarrow T^n$ whose image does not intersects a type of an orbit since the number of type is finite. $S^1$ acts freely on $M$ via $f$.
References Torus Actions on Symplectic Manifolds, Michele Audin.