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Quasi-actions and almost normal subgroups
Virtualconference
Authors : Margolis, Alex (Author of the conference)
CIRM (Publisher )
20E08 - Groups acting on trees
20F65 - Geometric group theory
20J05 - Homological methods in group theory
57M07 - Topological methods in group theory
Additional resources :
https://conferences.cirm-math.fr/uploads/1/6/6/4/16648158/pres.pdf
CIRM (Publisher )
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Abstract :
If a group G acts isometrically on a metric space X and Y is any metric space that is quasi-isometric to X, then G quasi-acts on Y. A fundamental problem in geometric group theory is to straighten (or quasi-conjugate) a quasi-action to an isometric action on a nice space. We will introduce and investigate discretisable spaces, those for which every cobounded quasi-action can be quasi-conjugated to an isometric action of a locally finite graph. Work of Mosher-Sageev-Whyte shows that free groups have this property, but it holds much more generally. For instance, we show that every hyperbolic group is either commensurable to a cocompact lattice in rank one Lie group, or it is discretisable.
We give several applications and indicate possible future directions of this ongoing work, particularly in showing that normal and almost normal subgroups are often preserved by quasi-isometries. For instance, we show that any finitely generated group quasi-isometric to a Z-by-hyperbolic group is Z-by-hyperbolic. We also show that within the class of residually finite groups, the class of central extensions of finitely generated abelian groups by hyperbolic groups is closed under quasi-isometries.
Keywords : Quasi-action; quasi-isometry; almost normal; hyperbolic
MSC Codes : We give several applications and indicate possible future directions of this ongoing work, particularly in showing that normal and almost normal subgroups are often preserved by quasi-isometries. For instance, we show that any finitely generated group quasi-isometric to a Z-by-hyperbolic group is Z-by-hyperbolic. We also show that within the class of residually finite groups, the class of central extensions of finitely generated abelian groups by hyperbolic groups is closed under quasi-isometries.
Keywords : Quasi-action; quasi-isometry; almost normal; hyperbolic
20E08 - Groups acting on trees
20F65 - Geometric group theory
20J05 - Homological methods in group theory
57M07 - Topological methods in group theory
Additional resources :
https://conferences.cirm-math.fr/uploads/1/6/6/4/16648158/pres.pdf
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Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 01/06/2020 Conference Date : 27/05/2020 Subseries : Research talks arXiv category : Geometric Topology ; Group Theory Mathematical Area(s) : Geometry ; Topology Format : MP4 (.mp4) - HD Video Time : 00:33:48 Targeted Audience : Researchers ; General Public Download : https://videos.cirm-math.fr/2020-05-27_Margolis.mp4 
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Information on the Event
Event Title : Virtual Geometric Group Theory conference / Rencontre virtuelle en géométrie des groupesEvent Organizers : Chatterji, Indira ; Paris, Luis ; Vogtmann, Karen Dates : 01/06/2020 - 05/06/2020 Event Year : 2020 Event URL : https://conferences.cirm-math.fr/virtual...
Citation Data
DOI : 10.24350/CIRM.V.19636503Cite this video as: Margolis, Alex (2020). Quasi-actions and almost normal subgroups. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19636503 URI : http://dx.doi.org/10.24350/CIRM.V.19636503 |
See Also
- [Virtualconference] Incoherence of free-by-free and surface-by-free groups / Author of the conference Walsh, Genevieve.
- [Virtualconference] Automorphism groups of Coxeter groups do not have Kazhdan's property (T) / Author of the conference Varghese, Olga.
- [Virtualconference] Action rigidity for free products of hyperbolic manifold groups / Author of the conference Stark, Emily.
- [Virtualconference] Intersections and joins of subgroups in free groups / Author of the conference Soroko, Ignat.
- [Virtualconference] Effectively generating RAAGs in MCGs / Author of the conference Runnels, Ian.
- [Virtualconference] Parabolic subgroups of infinite type Artin groups / Author of the conference Morris-Wright, Rose.
- [Virtualconference] Action of the Cremona group on a CAT(0) cube complex / Author of the conference Lonjou, Anne.
- [Virtualconference] Right-angled Coxeter groups commensurable to right-angled Artin groups / Author of the conference Levcovitz, Ivan.
- [Virtualconference] Computing fibring of 3-manifolds and free-by-cyclic groups / Author of the conference Kielak, Dawid.
- [Virtualconference] Shortcut graphs and groups / Author of the conference Hoda, Nima.
- [Virtualconference] Spin mapping class groups and curve graphs / Author of the conference Hamenstädt, Ursula.
- [Virtualconference] Spaces of cubulations / Author of the conference Fioravanti, Elia.
- [Virtualconference] Complexes of parabolic subgroups for Artin groups / Author of the conference Cumplido Cabello, Maria.
- [Virtualconference] Quasi-parabolic structures on groups / Author of the conference Balasubramanya, Sahana.
Bibliography
- KAPOVICH, Michael, KLEINER, Bruce, et LEEB, Bernhard. Quasi-isometries and the de Rham decomposition. Topology, 1998, vol. 37, no 6, p. 1193-1211. - https://doi.org/10.1016/S0040-9383(97)00091-8
- MOSHER, Lee, SAGEEV, Michah, et WHYTE, Kevin. Quasi-actions on trees I. Bounded valence. Annals of mathematics, 2003, p. 115-164. - https://www.jstor.org/stable/3597155
- MARGOLIS, Alexander. The geometry of groups containing almost normal subgroups. arXiv preprint arXiv:1905.03062, 2019. - https://arxiv.org/abs/1905.03062
- MARGOLIS, Alexander. Discretisable quasi-actions. in preparation (2020) -
