Let $G$ be a compact connected Lie group acting smoothly on a smooth connected manifold $M$. Say that $p,q\in M$ have the same orbit type if their stabilizer subgroups $G_p$ and $G_q$ are conjugate in $G$.

Can there be infinitely many orbit types?

In other words, if $[G_p]$ denotes the conjugacy class of $G_p$ in $G$, is it possible that the set $$\{[G_p]:p\in M\}$$ is infinite?

  • $\begingroup$ Is M connected? compact?.. $\endgroup$ – Tsemo Aristide Sep 20 '16 at 15:44
  • $\begingroup$ @TsemoAristide As stated in question, $M$ is connected. I don't assume that $M$ is compact, but if you have an answer in that situation, I would still be interested. $\endgroup$ – Simon Parker Sep 20 '16 at 15:45

There exits finitely orbit type if $M$ is compact. See p. 15 here



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.