The geometry of generalized loxodromic elements

Carolyn R. Abbott; David Hume

doi:10.5802/aif.3379

The geometry of generalized loxodromic elements

Carolyn R. Abbott, David Hume

Mathematics

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Abstract

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.

Original language	English
Pages (from-to)	1689-1713
Number of pages	25
Journal	Annales de l'Institut Fourier
Volume	70
Issue number	4
DOIs	https://doi.org/10.5802/aif.3379
Publication status	Published - 15 Apr 2021

Bibliographical note

Acknowledgments:
The first author was partially supported by the NSF RTG awards DMS-1502553. The second author was supported by the NSF grant DMS-1440140 while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2016 semester, and by a Titchmarsh Research Fellowship from the University of Oxford.

Keywords

hyperbolicity
acylindrical hyperbolicity

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title = "The geometry of generalized loxodromic elements",

abstract = "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{\textquoteright}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.",

keywords = "hyperbolicity, acylindrical hyperbolicity",

author = "Abbott, {Carolyn R.} and David Hume",

note = "Acknowledgments: The first author was partially supported by the NSF RTG awards DMS-1502553. The second author was supported by the NSF grant DMS-1440140 while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2016 semester, and by a Titchmarsh Research Fellowship from the University of Oxford.",

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doi = "10.5802/aif.3379",

language = "English",

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AU - Abbott, Carolyn R.

AU - Hume, David

N1 - Acknowledgments: The first author was partially supported by the NSF RTG awards DMS-1502553. The second author was supported by the NSF grant DMS-1440140 while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2016 semester, and by a Titchmarsh Research Fellowship from the University of Oxford.

PY - 2021/4/15

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N2 - 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.

AB - 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.

KW - hyperbolicity

KW - acylindrical hyperbolicity

U2 - 10.5802/aif.3379

DO - 10.5802/aif.3379

M3 - Article

SN - 1777-5310

VL - 70

SP - 1689

EP - 1713

JO - Annales de l'Institut Fourier

JF - Annales de l'Institut Fourier

IS - 4

ER -

The geometry of generalized loxodromic elements

Abstract

Bibliographical note

Keywords

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