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arXiv · 1411.5913

The converse of the Schwarz Lemma is false

Maxime Fortier Bourque
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Abstract

Let $h:X \to Y$ be a homeomorphism between hyperbolic surfaces with finite topology. If $h$ is homotopic to a holomorphic map, then every closed geodesic in $X$ is at least as long as the corresponding geodesic in $Y$, by the Schwarz Lemma. The converse holds trivially when $X$ and $Y$ are disks or annuli, and it holds when $X$ and $Y$ are closed surfaces by a theorem of W. Thurston. We prove that the converse is false in all other cases, strengthening a result of Masumoto.

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Maxime Fortier Bourque. 2014-11-21. The converse of the Schwarz Lemma is false. https://arxiv.org/abs/1411.5913

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