Abstract
We compare two families of left-invariant metrics on a surface group Γ = π1(Σ) in the context of course-geometry. One family comes from Riemannian metrics of negative curvature on the surface Σ, and another from quasi-Fuchsian representations of Γ. We show that the Teichmüller space \({\cal T}\) (Σ) is the only common part of these two families, even when viewed from the coarse-geometric perspective.
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Acknowledgements
This note is dedicated to Benjy Weiss on the occasion of his 80th birthday. His profound contributions to Ergodic Theory and Dynamics, breadth of his interests and originality of his ideas are an inspiration to us and many, many others.
We would like to thank Dick Canary for the useful discussion about quasi-Fuchsian representations.
A. F. acknowledges the support of the NSF grant DMS-2005493 and BSF grant-2018258.
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To Benjy Weiss with gratitude and admiration
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Fricker, E., Furman, A. Quasi-Fuchsian vs negative curvature metrics on surface groups. Isr. J. Math. 251, 365–378 (2022). https://doi.org/10.1007/s11856-022-2440-1
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DOI: https://doi.org/10.1007/s11856-022-2440-1