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LARGE-SCALE SUBLINEARLY LIPSCHITZ GEOMETRY OF HYPERBOLIC SPACES

Part of: Analysis on metric spaces Global differential geometry Special aspects of infinite or finite groups Geometric function theory

Published online by Cambridge University Press:  19 December 2018

Gabriel Pallier
Gabriel Pallier*
Affiliation:
Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405Orsay, France ([email protected])
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Abstract

Large-scale sublinearly Lipschitz maps have been introduced by Yves Cornulier in order to precisely state his theorems about asymptotic cones of Lie groups. In particular, Sublinearly bi-Lipschitz Equivalences (SBE) are a weak variant of quasi-isometries, with the only requirement of still inducing bi-Lipschitz maps at the level of asymptotic cones. We focus here on hyperbolic metric spaces and study properties of boundary extensions of SBEs, reminiscent of quasi-Möbius (or quasisymmetric) mappings. We give a dimensional invariant of the boundary that allows to distinguish hyperbolic symmetric spaces up to SBE, answering a question of Druţu.

Keywords

hyperbolic groupsquasiconformal mappingscross-ratiosasymptotic geometryanalysis on metric spaces

MSC classification

Primary: 20F67: Hyperbolic groups and nonpositively curved groups 30C65: Quasiconformal mappings in $^n$, other generalizations
Secondary: 53C23: Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces 20F69: Asymptotic properties of groups 30L10: Quasiconformal mappings in metric spaces
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 19 , Issue 6 , November 2020 , pp. 1831 - 1876
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Copyright
© Cambridge University Press 2018

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Footnotes

The author thanks the support of project ANR-15-CE40-0018.

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