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Forgive my ignorance on the Lie theory. I have the following questions in my current work concerning a certain property of compact connected Lie groups.

First, allow me to fix some notations. Let $G$ be a connected compact Lie group, $\mathfrak{g}$ its Lie algebra. Then $\operatorname{Aut}(G)$, the group of (smooth) automorphisms of the $G$, is a closed subgroup of $\operatorname{Aut}(\mathfrak{g})$, hence is a Lie group itself. Let $\operatorname{ad} : G \to \operatorname{Aut}(G) \subseteq \operatorname{Aut}(\mathfrak{g})$ be the adjoint representation and denote its image by $\operatorname{Inn}(G)$, and needless to say, $\operatorname{Inn}(G)$ is a normal subgroup of $\operatorname{Aut}(G)$. We denote the quotient group $\operatorname{Aut}(G) / \operatorname{Inn}(G)$ by $\operatorname{Out}(G)$, and this is what I mean when I say the group of outer automorphisms in the title.

Question A. When $\operatorname{Out}(G)$ is countable, is $\operatorname{Inn}(G)$ always open in $\operatorname{Aut}(G)$?

More generally, although this seems to have a higher chance to have a negative answer, one can ask the following question, which I suspect is still true:

Question B. Is $\operatorname{Inn}(G)$ always open in $\operatorname{Aut}(G)$? If Question A has an affirmative answer, this is equivalent to ask if $\operatorname{Out}(G)$ is always countable.

Question C. When $\operatorname{Inn}(G)$ is open in $\operatorname{Aut}(G)$, is $\operatorname{Out}(G)$ (which now a countable discrete group) always amenable as a locally compact group?

A little experiment on the abelian case of the $n$-dimensional torus (in which case the inner automorphism group is trivial, and $\operatorname{Aut}(T^n)$ is $GL_n(\mathbb{Z})$), and the semisimple case (at least over $\mathbb{C}$ instead of $\mathbb{R}$, which implies $\operatorname*{Out}(G)$ being finite), seems to make a negative answer to the above questions not completely trivial. Although I am happy that the abelian case and the semisimple case already suffice for the purpose of my current work, which only needs $\operatorname{Out}(G)$ to be amenable, I would be very interested if one can prove or disprove the amenability of $\operatorname*{Out}(G)$ for a general compact connected Lie group $G$.

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    $\begingroup$ It's not hard to check that the automorphism group of a compact connected is a Lie group, whose unit component consists exactly of the inner automorphisms. If $d$ is the dimension of the central torus, it always has a discrete subgroup isomorphic to a finite index subgroup of $\mathrm{GL}_d(\mathbf{Z})$, which maps injectively and with image in the outer automorphism group (the group of components of this Lie group). Hence $\mathrm{Aut}(G)$ is amenable $\Leftrightarrow$ the discrete group $\mathrm{Out}(G)$ is amenable $\Leftrightarrow$ $\mathrm{Out}(G)$ is finite $\Leftrightarrow$ $d\le 1$. $\endgroup$
    – YCor
    Commented Jun 27, 2019 at 9:42
  • $\begingroup$ @YCor Thanks for this nice comment, which seems to completely solved my questions. I am not very familiar with Lie theory. Could you please point some reference for me on the techniques for the arguments you gave? Mainly on the part of the existence of a discrete subgroup isomorphic to a finite index subgroup of $\operatorname{GL}_d(\mathbb{Z})$. $\endgroup$
    – Hua Wang
    Commented Jun 27, 2019 at 10:51
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    $\begingroup$ You should first try in the case of the direct product of a torus and a semisimple compact Lie group. $\endgroup$
    – YCor
    Commented Jun 27, 2019 at 12:05

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