Timeline for Has dynamics on $G/\Gamma$ ever been used to prove interesting things about $\Gamma$?
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Sep 29, 2018 at 0:44 | comment | added | Kim | @Asaf Feel free to restrict yourself to finite covolume if you like. I've formulated, a few comments back, the question in that setting as well. | |
| Sep 28, 2018 at 15:39 | comment | added | Asaf | @Venkataramana, the B-Q papers deal with (algebraicity of...) stationary measures generated by actions of subgroups (say discrete) over homogeneous spaces, but the homogeneous space itself is of finite volume! There are interesting works in the infinite volume setting (Oh et al. Sarig-Lederappier), but things are much less understood in those settings, and a lot of the techniques mentioned by Kim are unavailable. When some are do available, it usually involves brute forcing through intricate properties of the discrete subgroup that allow the finite-vol techniques to work. | |
| Sep 21, 2018 at 15:06 | history | made wiki | Post Made Community Wiki by S. Carnahan♦ | ||
| Sep 21, 2018 at 1:43 | comment | added | Venkataramana | papers of Benoit and Quint deal with this infinite covolume setting. | |
| Sep 20, 2018 at 22:07 | comment | added | Kim | That aside, let us assume for the moment that $\Gamma$ does arise as a lattice. Then my question is, what group-theoretic properties can we deduce about $\Gamma$ from homogeneous dynamics on $G/\Gamma$? | |
| Sep 20, 2018 at 21:06 | comment | added | Kim | Measure classification, orbit (closure) classification, equidistribution, non-divergence in various forms, etc. Also more mundane things such as descriptions of certain trajectories. Really anything that involves studying actions of $H$ on $G/\Gamma$. These can be formulated in the infinite volume setting. | |
| Sep 20, 2018 at 20:58 | comment | added | Asaf | What is "the homogeneous dynamics of $G/\Gamma$" without further assumptions on $\Gamma$? | |
| Sep 20, 2018 at 20:16 | comment | added | Kim | I'm not making any assumption about being a lattice. $\Gamma$ could well have infinite covolume. I am wondering what we can deduce about $\Gamma$ (as an abstract group) as a consequence of exploring the homogeneous dynamics of $G/\Gamma$. | |
| Sep 20, 2018 at 19:58 | comment | added | Asaf | The finite generation comes from say property T, this was the original motivation of Kazhdan and Margulis. Basically your question needs to be phrased as ''which groups arise as lattices'', and the rigidity theorems rather limits one to specific types of groups. | |
| Sep 20, 2018 at 19:43 | comment | added | Kim | What I mean is that I consider $\Gamma$ as an (isomorphism class of) abstract group. In principle I could embed $\Gamma$ as a lattice in some $G$, as a non-lattice in another $G'$, etc. So being a lattice is not an intrinsic property of $\Gamma$. Intrinsic questions would be things like: What are the normal subgroups of $\Gamma$? Is $\Gamma$ finitely generated? and so on. | |
| Sep 20, 2018 at 19:39 | comment | added | Asaf | Some of the proofs of Mostow rigidity for example would use mixing, but I still don't get what you mean by intrinsic, you begin by saying that $\Gamma$ is a lattice in $G$, that gives you quite a bit of information. | |
| Sep 20, 2018 at 19:35 | comment | added | Kim | These are all very interesting results, certainly. What I am looking for is the subset of results about intrinsic properties of $\Gamma$ (rather than its properties as a subgroup of $G$). | |
| Sep 20, 2018 at 19:19 | history | answered | Asaf | CC BY-SA 4.0 |