The geometry of generalized loxodromic elements
[La géométrie des éléments loxodromiques généralisés]
Annales de l'Institut Fourier, Tome 70 (2020) no. 4, pp. 1689-1713.

Nous présentons des conditions suffisantes pour qu’un élément d’un groupe de type fini soit, ou ne soit pas, loxodromique généralisé ; dans ce cadre, nous prouvons une généralisation du résultat de Sisto selon lequel tout élément loxodromique généralisé a la propriété de Morse. Nous donnons une condition géométrique suffisante pour qu’un élément d’un groupe de petite simplification soit loxodromique généralisé en termes des relations définissant le groupe et fournissons plusieurs constructions prouvant que cette condition est optimale.

We explore geometric conditions which ensure that a given element of a finitely generated group is, or fails to be, generalized loxodromic; as part of this we prove a generalization of Sisto’s result that every generalized loxodromic element is Morse. We provide a sufficient geometric condition for an element of a small cancellation group to be generalized loxodromic in terms of the defining relations and provide a number of constructions which prove that this condition is sharp.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3379
Classification : 20F65, 20F05, 20F06
Keywords: hyperbolicity, acylindrical hyperbolicity, small cancellation.
Mot clés : hyperbolicité, hyperbolicité acylindrique, petite simplification.
Abbott, Carolyn R. 1 ; Hume, David 2

1 Department of Mathematics, Columbia University, 2990 Broadway, New York, NY 10027 (USA)
2 Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG (United Kingdom)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits
@article{AIF_2020__70_4_1689_0,
     author = {Abbott, Carolyn R. and Hume, David},
     title = {The geometry of generalized loxodromic elements},
     journal = {Annales de l'Institut Fourier},
     pages = {1689--1713},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {70},
     number = {4},
     year = {2020},
     doi = {10.5802/aif.3379},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3379/}
}
TY  - JOUR
AU  - Abbott, Carolyn R.
AU  - Hume, David
TI  - The geometry of generalized loxodromic elements
JO  - Annales de l'Institut Fourier
PY  - 2020
SP  - 1689
EP  - 1713
VL  - 70
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3379/
DO  - 10.5802/aif.3379
LA  - en
ID  - AIF_2020__70_4_1689_0
ER  - 
%0 Journal Article
%A Abbott, Carolyn R.
%A Hume, David
%T The geometry of generalized loxodromic elements
%J Annales de l'Institut Fourier
%D 2020
%P 1689-1713
%V 70
%N 4
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3379/
%R 10.5802/aif.3379
%G en
%F AIF_2020__70_4_1689_0
Abbott, Carolyn R.; Hume, David. The geometry of generalized loxodromic elements. Annales de l'Institut Fourier, Tome 70 (2020) no. 4, pp. 1689-1713. doi : 10.5802/aif.3379. https://aif.centre-mersenne.org/articles/10.5802/aif.3379/

[1] Abbott, Carolyn R. Not all finitely generated groups have universal acylindrical actions, Proc. Am. Math. Soc., Volume 144 (2016) no. 10, pp. 4151-4155 | DOI | MR | Zbl

[2] Abbott, Carolyn R.; Behrstock, Jason; Durham, Matthew Gentry Largest acylindrical actions and stability in hierarchically hyperbolic groups (2017) (https://arxiv.org/abs/1705.06219) | Zbl

[3] Arzhantseva, Goulnara N.; Cashen, Christopher H.; Gruber, Dominik; Hume, David Characterizations of Morse quasi-geodesics via superlinear divergence and sublinear contraction, Doc. Math., Volume 22 (2017), pp. 1193-1224 | MR | Zbl

[4] Arzhantseva, Goulnara N.; Cashen, Christopher H.; Gruber, Dominik; Hume, David Negative curvature in graphical small cancellation groups, Groups Geom. Dyn., Volume 13 (2019) no. 2, pp. 579-632 | DOI | MR | Zbl

[5] Bestvina, Mladen; Bromberg, Ken; Fujiwara, Koji Constructing group actions on quasi-trees and applications to mapping class groups, Publ. Math., Inst. Hautes Étud. Sci., Volume 122 (2015), pp. 1-64 | DOI | MR | Zbl

[6] Bestvina, Mladen; Fujiwara, Koji Bounded cohomology of subgroups of mapping class groups, Geom. Topol., Volume 6 (2002), pp. 69-89 | DOI | MR | Zbl

[7] Bowditch, Brian H. Continuously many quasi-isometry classes of 2-generator groups, Comment. Math. Helv., Volume 73 (1998) no. 2, pp. 232-236 | DOI | MR | Zbl

[8] geodesics in the curve complex, Tight Bowditch, Brian H., Invent. Math., Volume 171 (2008) no. 2, pp. 281-300 | Zbl

[9] Coulon, Rémi; Gruber, Dominik Small cancellation theory over Burnside groups (2017) (https://arxiv.org/abs/1705.09651) | DOI | Zbl

[10] Dahmani, François; Guirardel, Vincent; Osin, Denis V. Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces, Mem. Am. Math. Soc., Volume 245 (2017) no. 1156, p. v+152 | DOI | MR | Zbl

[11] Gruber, Dominik Infinitely presented C(6)-groups are SQ-universal, J. Lond. Math. Soc., Volume 92 (2015) no. 1, pp. 178-201 | DOI | MR | Zbl

[12] Gruber, Dominik; Sisto, Alessandro Infinitely presented graphical small cancellation groups are acylindrically hyperbolic, Ann. Inst. Fourier, Volume 68 (2018) no. 6, pp. 2501-2552 | DOI | Numdam | MR | Zbl

[13] Hamenstädt, Ursula Bounded cohomology and isometry groups of hyperbolic spaces, J. Eur. Math. Soc., Volume 10 (2008) no. 2, pp. 315-349 | DOI | MR | Zbl

[14] Hull, Michael Small cancellation in acylindrically hyperbolic groups, Groups Geom. Dyn., Volume 10 (2016) no. 4, pp. 1077-1119 | DOI | MR | Zbl

[15] Osin, Denis V. Acylindrically hyperbolic groups, Trans. Am. Math. Soc., Volume 368 (2016) no. 2, pp. 851-888 | DOI | MR | Zbl

[16] Sisto, Alessandro Quasi-convexity of hyperbolically embedded subgroups, Math. Z., Volume 283 (2016) no. 3-4, pp. 649-658 | DOI | MR | Zbl

Cité par Sources :