Let $G$ be a connected second countable compact Hausdorff group, and let $H\subset G$ be a closed subgroup such that the homogeneous space $G/H$ has continuum cardinality. For every $x\in G/H$ let $H_x\subset G$ be the stability subgroup at $x$.

Question: Does there exist a toral subgroup $\mathbb{T}\subset G$ such that the subset $\{x\in G/H\,|\quad\mathbb{T}\cap H_x\neq\{1\}\}$ is at most countable?

Here $\mathbb{T}\subset G$ is a closed subgroup isomorphic to the usual torus $U(1)$.

This may turn out to be a very easy question, I just can't figure out an efficient approach. Thank you.

  • 2
    $\begingroup$ Some questions : what do you mean by toral subgroup ? abelian connected subgroup ? compact ? The intersection $T\cap H_x$ contains the neutral element, you may not want to require the intersection to be empty ? $\endgroup$ – user120527 Feb 16 '18 at 12:13
  • 1
    $\begingroup$ Yes, what is "a toral subgroup"? Note: The Pontryagin dual of the rationals doesn't have any subgroup isomorphic to $S^1$. $\endgroup$ – Uri Bader Feb 16 '18 at 13:45
  • $\begingroup$ Rationals with which topology? Discrete? Is the dual connected in that case? Note that I explicitly required $G$ to be connected. Every pro-torus in a connected compact group is a projective limit of finite dimensional tori, and in particular contain a 1-torus. $\endgroup$ – Bedovlat Feb 16 '18 at 13:53
  • $\begingroup$ Let $G$ be the solenoid and $H=\{1\}$. Then $G$ contains no total subgroups. $\endgroup$ – Misha Feb 16 '18 at 14:53
  • $\begingroup$ I just realized Uri Bader mentioned the same example since the solenoid is isomorphic to the Pontryagin dual of the additive group of rational numbers (with the standard topology). $\endgroup$ – Misha Feb 16 '18 at 15:13

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.