Timeline for How does a quasi-isometry affect Poisson or Martin boundaries?
Current License: CC BY-SA 3.0
16 events
| when toggle format | what | by | license | comment | |
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| Mar 1, 2013 at 16:26 | comment | added | R W | @Igor Khavkine - Your comment about comparison of Martin and Busemann compactifications is only valid for symmetric spaces. | |
| Sep 15, 2012 at 16:55 | vote | accept | ARG | ||
| Sep 14, 2012 at 1:01 | comment | added | Misha | @Igor: OK, here it is. | |
| Sep 14, 2012 at 1:00 | answer | added | Misha | timeline score: 7 | |
| Sep 13, 2012 at 21:59 | answer | added | Ori Gurel-Gurevich | timeline score: 5 | |
| Sep 13, 2012 at 20:34 | comment | added | Igor Khavkine | @Misha, it seems like you have counterexamples, which then answers the question. How about collecting your comments in an answer? | |
| Sep 13, 2012 at 20:00 | comment | added | Misha | The first example to consider is the class of maps $F(x,y)=(x, y+f(x))$, where $f$ is Lipschitz. Then $F$ is bilipschitz. Look what it does to the x-axis. You can do the same in polar coordinates to get even worse behavior. | |
| Sep 13, 2012 at 12:08 | comment | added | Misha | The reference is Terry Lyons paper in Journal of Diff Geometry, 1987. Igor: Geodesics do not map close to geodesics unless you are in Gromov hyperbolic setting. This fails already for Euclidean plane. Moreover, there examples of Croke and Kleiner where topological type of CAT(0) boundary is not preserved by quasi-isometries. | |
| Sep 13, 2012 at 11:46 | comment | added | ARG | @Misha: thanks, could you point to a reference? But could one interpret this as the fact that the Poisson measure collapse to a Dirac mass? @Igor: indeed, that seems to be the proper question at the moment. | |
| Sep 13, 2012 at 11:38 | comment | added | Igor Khavkine | It seems to me that quasi-isometries would preserve equivalence classes of curves with "eventually bounded distance" (formula in my first comment). Equivalence classes that contain geodesics constitute the geodesic boundary. Are geodesic classes bijectively mapped to geodesic classes? | |
| Sep 13, 2012 at 11:31 | comment | added | Igor Khavkine | @Misha, I think that is less of an obstacle if the Martin boundary is constructed using positive superharmonic functions. But that is an issue for the precise set of hypothesis that the OP desires. | |
| Sep 13, 2012 at 11:08 | comment | added | Misha | Even existence of nonconstant positive harmonic functions is not invariant under quasi-isometries. | |
| Sep 13, 2012 at 10:06 | comment | added | ARG | Thanks for the reference in any case! A quasi-isometry would send those two geodesics at bounded distance but they might no longer be geodesics. Perhaps there are examples where the geodesic boundary is not preserve... I'll have to look into this. | |
| Sep 13, 2012 at 10:02 | history | edited | ARG | CC BY-SA 3.0 |
added definition
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| Sep 13, 2012 at 9:13 | comment | added | Igor Khavkine | Not an answer, unfortunately. It is known that the Martin boundary dominates the geodesic boundary (Prop.I.7.15 of Borel & Lizhen, Compactifications of symmetric and locally symmetric spaces). So if quasi-isometry does not preserve the geodesic boundary then it does not preserve the Martin one either. Points of the geodesic boundary are equivalence classes of (unit speed, I think) geodesics $\gamma(t)$ such that $\limsup_{t\to\infty} d(\gamma_1(t),\gamma_2(t)) < +\infty$. How does a quasi-isometry act on geodesics? | |
| Sep 13, 2012 at 8:20 | history | asked | ARG | CC BY-SA 3.0 |