Let $G$ be a simply-connected compact topological group (you can think of $SU(n)$ if you like it more concrete), and let $X$ be a finite-dimensional simply-connected $G$-CW-complex. If we know that all the isotropy groups are finite, does this imply they are trivial? And if not, are they at least bounded in size (in some sense)?



Almost free actions on simply-connected spaces
Since it was requested, here is the answer (from the comment) again.


The statement is wrong. For each $k$, one can start with a unique $0$-cell $G/(\mathbb Z/k)$ and attach a $2$-cell $G \times D^2$ by a map $G \times S^1 \to G/(\mathbb Z/k)$ that equivariantly extends a generator $S^1 \to G/(\mathbb Z/k)$ of the fundamental group $\pi\_1(G/(\mathbb Z/k)) = \mathbb Z/k$. The resulting $G$-CW complex will be simply-connected, with finite, but non-trivial isotropy subgroups.


