Annales Academiæ Scientiarum Fennicæ

Mathematica

Volumen 41, 2016, 235-241

# THE CONVERSE OF THE SCHWARZ LEMMA IS FALSE

## Maxime Fortier Bourque

University of Toronto, Department of Mathematics

40 St. George Street, Toronto, Ontario, Canada M5S 2E4;
mbourque 'at' math.toronto.edu

**Abstract.**
Let *h* : *X* → *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 Thurston. We prove that the converse is
false in all other cases, strengthening a result of Masumoto.

**2010 Mathematics Subject Classification:**
Primary 30F45, 30F60, 32G15.

**Key words:**
Schwarz Lemma, hyperbolic surfaces,
length of geodesics.

**Reference to this article:** M. Fortier Bourque:
The converse of the Schwarz Lemma is false.
Ann. Acad. Sci. Fenn. Math. 41 (2016), 235-241.

Full document as PDF file

doi:10.5186/aasfm.2016.4115

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