Timeline for Continuity of the Green function with respect to the measure
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
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| Jul 24, 2019 at 12:16 | comment | added | M. Dus | @MateuszKwaśnicki Thanks a lot. Discussing it with a friend, we've made some progress. Proposition 6.3 in Mathieu and Sisto's paper (arxiv.org/abs/1411.7865) shows Lipschitz continuity of the Green distance $d_G(e,x)$ with respect to both $\mu$ and the word distance $d(e,x)$. One might be able to adapt their proof to show Lipschitz continuity of $G(\mu)$ with respect to $\mu$. However, they assume that (a) the group is non-amenable and (b) $\mu$ is a probability measure. In particular, it seems it could not be applied to $r\mu$, where $r$ is below the spectral radius of $\mu$... | |
| Jul 18, 2019 at 17:42 | comment | added | Mateusz Kwaśnicki | Since this has not received any answers yet, let me spell it out that (a) I find the question quite interesting, (b) by Fatou's lemma, $G(\mu)(g)$ is clearly lower semi-continuous; (c) I fail to find a counter-example. | |
| Jul 12, 2019 at 22:10 | comment | added | M. Dus | @MateuszKwaśnicki Well I had implicitly in mind a non-negative measure, but you're right it might not change the result nor the proof | |
| Jul 12, 2019 at 10:57 | comment | added | Mateusz Kwaśnicki | Is $\mu$ a signed measure, or a non-negative measure? (Not sure if this affects the answer in any way, just asking.) | |
| Jul 9, 2019 at 16:59 | history | asked | M. Dus | CC BY-SA 4.0 |