# Amenability of the group of outer automorphisms of a connected compact Lie group

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 $$G$$.

• 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 $\Leftrightarrow$ $d\le 1$.
– YCor
Commented Jun 27, 2019 at 9:42
• @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 $\operatorname{GL}_d(\mathbb{Z})$. Commented Jun 27, 2019 at 10:51
• You should first try in the case of the direct product of a torus and a semisimple compact Lie group.
– YCor
Commented Jun 27, 2019 at 12:05