Induced quasi-isometries of hyperbolic spaces, Markov chains, and acylindrical hyperbolicity

Antoine Goldsborough; Mark Hagen; Harry Petyt; Alessandro Sisto; Jacob Russell

Induced quasi-isometries of hyperbolic spaces, Markov chains, and acylindrical hyperbolicity

Antoine Goldsborough, Mark Hagen, Harry Petyt, Alessandro Sisto, Jacob Russell

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Abstract

We show that quasi-isometries of (well-behaved) hierarchically hyperbolic groups descend to quasi-isometries of their maximal hyperbolic space. This has two applications, one relating to quasi-isometry invariance of acylindrical hyperbolicity, and the other a linear progress result for Markov chains.

The appendix, by Jacob Russel, contains a partial converse under the (necessary) condition that the maximal hyperbolic space is one-ended.

Original language	English
Journal	Groups, Geometry, and Dynamics
Publication status	Accepted/In press - 6 Jan 2025

Keywords

Mathematics - Group Theory
Mathematics - Probability

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title = "Induced quasi-isometries of hyperbolic spaces, Markov chains, and acylindrical hyperbolicity",

abstract = "We show that quasi-isometries of (well-behaved) hierarchically hyperbolic groups descend to quasi-isometries of their maximal hyperbolic space. This has two applications, one relating to quasi-isometry invariance of acylindrical hyperbolicity, and the other a linear progress result for Markov chains.The appendix, by Jacob Russel, contains a partial converse under the (necessary) condition that the maximal hyperbolic space is one-ended.",

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T1 - Induced quasi-isometries of hyperbolic spaces, Markov chains, and acylindrical hyperbolicity

AU - Goldsborough, Antoine

AU - Hagen, Mark

AU - Petyt, Harry

AU - Sisto, Alessandro

AU - Russell, Jacob

PY - 2025/1/6

Y1 - 2025/1/6

N2 - We show that quasi-isometries of (well-behaved) hierarchically hyperbolic groups descend to quasi-isometries of their maximal hyperbolic space. This has two applications, one relating to quasi-isometry invariance of acylindrical hyperbolicity, and the other a linear progress result for Markov chains.The appendix, by Jacob Russel, contains a partial converse under the (necessary) condition that the maximal hyperbolic space is one-ended.

AB - We show that quasi-isometries of (well-behaved) hierarchically hyperbolic groups descend to quasi-isometries of their maximal hyperbolic space. This has two applications, one relating to quasi-isometry invariance of acylindrical hyperbolicity, and the other a linear progress result for Markov chains.The appendix, by Jacob Russel, contains a partial converse under the (necessary) condition that the maximal hyperbolic space is one-ended.

KW - Mathematics - Group Theory

KW - Mathematics - Probability

M3 - Article

SN - 1661-7207

JO - Groups, Geometry, and Dynamics

JF - Groups, Geometry, and Dynamics

ER -

Induced quasi-isometries of hyperbolic spaces, Markov chains, and acylindrical hyperbolicity

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