A bounded homogeneous space which fails to be symmetric? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T08:53:36Z http://mathoverflow.net/feeds/question/50221 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/50221/a-bounded-homogeneous-space-which-fails-to-be-symmetric A bounded homogeneous space which fails to be symmetric? Hugo Chapdelaine 2010-12-23T03:14:15Z 2012-02-19T17:50:39Z <p>Do we have examples of a contractible <strong>bounded</strong> open set $D\subseteq\mathbf{C}^n$ such that $Hol(D)$ (the group of biholomorphisms $f:D\rightarrow D$) acts transitively on $D$ but such that there exists no symmetry at a given point $x\in D$ (so at all points by homogeneity). By a symmetry at $x$ I mean an element $s\in Hol(D)$ such that in a small neighborhood of $x$ only $x$ is fixed and $s^2=1$. </p> http://mathoverflow.net/questions/50221/a-bounded-homogeneous-space-which-fails-to-be-symmetric/50222#50222 Answer by Richard Borcherds for A bounded homogeneous space which fails to be symmetric? Richard Borcherds 2010-12-23T03:41:09Z 2010-12-23T03:41:09Z <p>There are some examples given by Pjateckiĭ-Šapiro in Classification of bounded homogeneous regions in n-dimensional complex space. Dokl. Akad. Nauk SSSR 141 1961 316–319. and On bounded homogeneous domains in an n-dimensional complex space. Izv. Akad. Nauk SSSR Ser. Mat. 26 1962 107–124. </p> <p>(I have a vague memory that the smallest examples are 4-dimensional, but might have misremembered.)</p> http://mathoverflow.net/questions/50221/a-bounded-homogeneous-space-which-fails-to-be-symmetric/88883#88883 Answer by Sigurdur Helgason for A bounded homogeneous space which fails to be symmetric? Sigurdur Helgason 2012-02-19T02:33:56Z 2012-02-19T02:33:56Z <p>E.Cartan proved in 1936 that for dimension 1 and 2 bounded homogeneous spaces are symmetric. For dimension 3 he did not publish the proof considering it loo long in comparison to the interest of the result. This has now changed with P-Sapiro's example for dimension 4. So the proof for dimension 3 is presumably somewhere in E.Cartan's Nachlass, unpublished.</p>