Timeline for Groups acting non-properly cocompactly on hyperbolic spaces
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 1 at 19:06 | answer | added | Lee Mosher | timeline score: 1 | |
| Nov 30 at 16:35 | answer | added | Ian Agol | timeline score: 1 | |
| Nov 30 at 11:19 | comment | added | YCor | Note: one gets zillions of examples by taking, e.g., a dense subgroup of $\mathrm{PSL}_2(\mathbf{R})$ acting on the hyperbolic plane, or similar. Also, for an arbitrary amalgam $A\ast_C B$, the action on the Bass-Serre tree is vertex-transitive, hence cobounded. But it's proper only when $A,B$ are both finite. | |
| Nov 30 at 11:16 | comment | added | YCor | Note: in my previous comment I was interpreting "cocompact" as "cobounded". Probably "cobounded" is the right assumption (rather than trying to specify the meaning of "cocompact" for non-proper actions). However, whether one assumes properness of the space should change the picture. | |
| Nov 30 at 10:06 | answer | added | AGenevois | timeline score: 4 | |
| Nov 30 at 9:05 | answer | added | ADL | timeline score: 3 | |
| Nov 30 at 6:31 | comment | added | Ian Agol | Other trivial example: take the (finite metric) cone over the Cayley graph of G. This is bounded diameter and a hyperbolic geodesic metric space, and G acts faithfully. Presumably the question needs some modifications to rule out bounded actions of this sort. Relatively hyperbolic groups have actions as well (coning off the peripheral subgroups) on non-locally compact spaces. One might be able to say something for actions on locally compact geodesic hyperbolic spaces with unbounded orbits? | |
| Nov 30 at 0:21 | comment | added | Matt Zaremsky | For non-proper cobounded actions on hyperbolic spaces, this is pretty much exactly the purview of Abbott--Balasubramanya--Osin's paper arxiv.org/abs/1710.05197 (and the many papers referencing it). Cocompact is more restrictive than cobounded, and I'm less sure of a good reference for a general picture of such things. (By the way a silly answer to your explicit question is, take any group you want and act trivially on a point.) | |
| Nov 30 at 0:20 | comment | added | YCor | Of course there are tons of such groups (e.g. direct product of hyperbolic with anything infinite). Beyond this, natural examples are lattices in products $X\times Y$ with $X$ hyperbolic. For an explicit example, consider for $n\ge 3$ the quadratic form $q=x_1^2+\dots+x_{n-2}^2+\sqrt{2}x_{n-1}^2-x_n^2$. Then $\mathrm{SO}(q)(\mathbf{Z}[\sqrt{2}]$ is a lattice in $\mathrm{SO}(n-1,1)\times\mathrm{SO}(n-2,2)$ and projects densely on $\mathrm{SO}(n-1,1)$, whence a cocompact but nonproper action. Also for $q'=x_1^2+\dots+x_{n-1}^2-x_n^2$, one gets an irreducible lattice in $\mathrm{SO}(n-1,1)^2$. | |
| Nov 29 at 23:52 | comment | added | bishop1989 | Yes! I have edited the question. Thanks. | |
| Nov 29 at 23:52 | history | edited | bishop1989 | CC BY-SA 4.0 |
added 18 characters in body
|
| Nov 29 at 23:48 | comment | added | YCor | It is important here that you specify whether you require the hyperbolic metric space to be geodesic. | |
| Nov 29 at 23:29 | history | asked | bishop1989 | CC BY-SA 4.0 |