no. 4
The geometry of generalized loxodromic elements
[La géométrie des éléments loxodromiques généralisés]
Abbott, Carolyn R.1 ; Hume, David2
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)
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.

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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 
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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/

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