Uniform undistortion from barycentres, and applications to hierarchically hyperbolic groups

Carolyn Abbott; Mark Hagen; Harry Petyt; Abdul Zalloum

doi:10.1515/agms-2025-0027

Uniform undistortion from barycentres, and applications to hierarchically hyperbolic groups

Carolyn Abbott
, Mark Hagen
, Harry Petyt^*
, Abdul Zalloum

^*Corresponding author for this work

School of Mathematics

Research output: Contribution to journal › Article (Academic Journal) › peer-review

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Abstract

We show that infinite cyclic subgroups of groups acting uniformly properly on injective metric spaces are uniformly undistorted. In the special case of hierarchically hyperbolic groups, we use this to study translation lengths for actions on the associated hyperbolic spaces. We then use quasimorphisms to produce examples where these latter results are sharp.

Original language	English
Article number	20250027
Number of pages	29
Journal	Analysis and Geometry in Metric Spaces
Volume	13
Issue number	1
DOIs	https://doi.org/10.1515/agms-2025-0027
Publication status	Published - 26 Nov 2025

Bibliographical note

Publisher Copyright:
© 2025 the author(s), published by De Gruyter.

Keywords

central extension
hierarchical hyperbolicity
injective space
translation length
undistorted

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title = "Uniform undistortion from barycentres, and applications to hierarchically hyperbolic groups",

abstract = "We show that infinite cyclic subgroups of groups acting uniformly properly on injective metric spaces are uniformly undistorted. In the special case of hierarchically hyperbolic groups, we use this to study translation lengths for actions on the associated hyperbolic spaces. We then use quasimorphisms to produce examples where these latter results are sharp.",

keywords = "central extension, hierarchical hyperbolicity, injective space, translation length, undistorted",

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AU - Abbott, Carolyn

AU - Hagen, Mark

AU - Petyt, Harry

AU - Zalloum, Abdul

PY - 2025/11/26

Y1 - 2025/11/26

N2 - We show that infinite cyclic subgroups of groups acting uniformly properly on injective metric spaces are uniformly undistorted. In the special case of hierarchically hyperbolic groups, we use this to study translation lengths for actions on the associated hyperbolic spaces. We then use quasimorphisms to produce examples where these latter results are sharp.

AB - We show that infinite cyclic subgroups of groups acting uniformly properly on injective metric spaces are uniformly undistorted. In the special case of hierarchically hyperbolic groups, we use this to study translation lengths for actions on the associated hyperbolic spaces. We then use quasimorphisms to produce examples where these latter results are sharp.

KW - central extension

KW - hierarchical hyperbolicity

KW - injective space

KW - translation length

KW - undistorted

U2 - 10.1515/agms-2025-0027

DO - 10.1515/agms-2025-0027

M3 - Article (Academic Journal)

SN - 2299-3274

VL - 13

JO - Analysis and Geometry in Metric Spaces

JF - Analysis and Geometry in Metric Spaces

IS - 1

M1 - 20250027

ER -

Uniform undistortion from barycentres, and applications to hierarchically hyperbolic groups

Abstract

Bibliographical note

Keywords

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