Timeline for Does a compact contractible metric space have a point that is fixed by all isometries?
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| May 14, 2022 at 10:45 | comment | added | Igor Belegradek | @YCor I only mentioned Mostow-Palais because the above also answers mathoverflow.net/questions/422384/…. | |
| May 14, 2022 at 7:51 | comment | added | YCor | Of course you don't need Mostow-Palais to get an invariant metric here: just take the standard Euclidean metric and average it along the smooth finite group action to get an invariant Riemannian metric. | |
| May 13, 2022 at 22:32 | comment | added | Igor Belegradek | @Gro-Tsen Yes disk=ball. Any compact topological group action on a 2-disk is equivalent to a linear action (hence has a fixed point). Any smooth action of a compact Lie group on the 3-disk is smoothly equivalent to a linear action. Any smooth finite group action on a 4-disk has a fixed point. For $n>5$ there is a fixed point free smooth action of the alternating group $A_5$ on the n-disk. I think the case $n=5$ is open. References can be found e.g. in the introduction and appendix A of arxiv.org/pdf/1706.08135.pdf. | |
| May 13, 2022 at 22:22 | comment | added | Gro-Tsen | By “a disk” you mean a ball, right? What about the question of whether a metric space homeomorphic a ($2$-dimensional closed) disk has a point fixed by all isometries? | |
| May 13, 2022 at 22:20 | vote | accept | M. Winter | ||
| May 13, 2022 at 18:21 | comment | added | Saúl RM | Other way to arrange the metric so that the action is isometric is, if $G$ is the group and $d$ is the usual metric in the disk, define $d'(x,y)=\sum_{g\in G}d(gx,gy)$ | |
| May 13, 2022 at 18:14 | history | answered | Igor Belegradek | CC BY-SA 4.0 |