Timeline for Non-zero homotopy/homology in diffeomorphism groups
Current License: CC BY-SA 3.0
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 14, 2018 at 14:51 | comment | added | Thomas Rot | Thank you for this very interesting class of examples. This is very surprising to me, which probably shows my ignorance of this topic. | |
| Feb 13, 2018 at 21:36 | comment | added | Igor Belegradek | There are certainly hyperbolic $3$-manifolds with trivial isometry group, and in fact, for each $n>1$ every finite group is the isometry group of hyperbolic $n$-manifold by a theorem of M.Belolipetsky and A.Lubotzky, see arxiv.org/pdf/math/0406607.pdf. The 3d case is due to S.Kojima. | |
| Feb 13, 2018 at 19:27 | comment | added | Ryan Budney | I don't believe there is a proof, but I suspect it's widely believed that "most" hyperbolic 3-manifolds have trivial symmetry group. Exactly what definition of "most" one uses could perhaps change the answer. But if you look at the ratio of hyperbolic 3-manifolds with symmetry to ones without, given a bound on the volume (as the volume goes to infinity) this should be zero. If it doesn't take long I'll look through SnapPy's census. . . | |
| Feb 13, 2018 at 19:10 | history | answered | Allen Hatcher | CC BY-SA 3.0 |