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Let $G$ be a domain in $\mathbb{C}^{n}$ and let $Aut(G)$ be the group of biholomorphic selfmaps of $G$. $G$ is called:

(1) homogeneous if $Aut(G)$ acts transitively on $G$, i.e. for any $z,w\in G$ there is $\varphi\in Aut(G)$, such that $\varphi(z)=w$;

(2z) symmetric at $z\in G$ if there is $\varphi\in Aut(G)$, such that $\varphi\circ\varphi =id$ and $z$ is an isolated fixed point of $\varphi$;

(2) symmetric if it is symmetric at any of its points.

It seems that (2) is equivalent to (1) and (2z) for some $z\in G$, but in many cases (1) alone implies (2).

Q1: Is there an elementary proof, that (2) implies (1)?

Additionally consider the following two properties:

(3z) for any $w\in G$ there there is $\varphi\in Aut(G)$, such that $\varphi(z)=w$ and $\varphi(w)=z$;

(3) for any $z,w\in G$ there there is $\varphi\in Aut(G)$, such that $\varphi(z)=w$ and $\varphi(w)=z$.

Clearly, (3z) for some $z\in G$ implies (1), and then using (1) in addition to (3z), one can show (3).

Also, using the classification of the bounded symmetric domains, one can show that (2) implies (3).

Q2: Does the property (3) have a name? Was it considered in the literature? Does (1) imply (3) back, or does (3z) follow from (2z) for some $z\in G$? Is there an elementary proof that (2) implies (3)?

Thank you.

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  • $\begingroup$ If by "elementary" you allow the introduction of the so called Bergman metric and basic facts of Riemannian geometry then all your questions had elementary answers. $\endgroup$
    – Holonomia
    Mar 23, 2017 at 10:30
  • $\begingroup$ @Holonomia thank you very much for your attention to my questions! However, it would be much more appreciated, if you posted a bit more elaborated answers. Yes, Bergman metric and its properties are on the edge of elementary for me (especially with references). $\endgroup$
    – erz
    Mar 23, 2017 at 20:58
  • $\begingroup$ More elaborated answers implies more time and unfortunately I have not much time. Anyway, for 2) implies 3) take the geodesic $\gamma$ whose with endpoints are $z,w$ and use the symmetry about the middle point of $\gamma$. Of course you need to show that the symmetry is an isometry and that the geodesic do exist. $\endgroup$
    – Holonomia
    Mar 23, 2017 at 21:22
  • $\begingroup$ @Holonomia Yes, I thought about this, but I don't understand why that can be done globally, not just in the neighbourhoods, where the exponential mapping is defined. $\endgroup$
    – erz
    Mar 23, 2017 at 21:29
  • $\begingroup$ Also, I am not sure if we can actually use Bergman metric for unbounded domains. $\endgroup$
    – erz
    Mar 23, 2017 at 21:45

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