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.
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.
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Accepted:
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Keywords: hyperbolicity, acylindrical hyperbolicity, small cancellation.
Mot clés : hyperbolicité, hyperbolicité acylindrique, petite simplification.
Abbott, Carolyn R. 1; Hume, David 2
@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, Volume 70 (2020) no. 4, pp. 1689-1713. doi : 10.5802/aif.3379. https://aif.centre-mersenne.org/articles/10.5802/aif.3379/
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