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The
Manhattan curve, ergodic theory of topological flows and
rigidity
Stephen Cantrell and Ryokichi Tanaka |
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Geometry & Topology 29 (2025) 1851–1907
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Abstract |
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For every nonelementary hyperbolic group, we introduce the Manhattan curve associated to each pair of left-invariant hyperbolic metrics which are quasi-isometric to a word metric. It is convex; we show that it is continuously differentiable and moreover is a straight line if and only if the corresponding two metrics are roughly similar, ie they are within bounded distance after multiplying by a positive constant. Further, we prove that the Manhattan curve associated to two strongly hyperbolic metrics is twice continuously differentiable. The proof is based on the ergodic theory of topological flows associated to general hyperbolic groups and analyzing the multifractal structure of Patterson–Sullivan measures. We exhibit some explicit examples including a hyperbolic triangle group and compute the exact value of the mean distortion for pairs of word metrics. |
Keywords
hyperbolic group, the Manhattan curve, Patterson–Sullivan
measure, symbolic dynamics, Hausdorff dimension
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Mathematical Subject Classification
Primary: 20F67
Secondary: 37D35, 37D40
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References |
Publication
Received: 9 November 2021
Revised: 23 September 2023
Accepted: 28 June 2024
Published: 27 June 2025
Proposed: Anna Wienhard
Seconded: Ian Agol, Dmitri Burago
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Authors |
| Stephen Cantrell | |
| Department of Mathematics University of Warwick Coventry United Kingdom |
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| Ryokichi Tanaka | |
| Department of Mathematics Kyoto University Kyoto Japan |
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| © 2025 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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