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Journal of the London Mathematical Society Advance Access published online on December 13, 2007
Journal of the London Mathematical Society, doi:10.1112/jlms/jdm100
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© 2007 London Mathematical Society

A geometric characterization of orientation-reversing involutions

Antonio F. Costa

Departmento de Matemáticas Fundamentales
Facultad de Ciencias
UNED
Madrid 28040
Spain
acosta{at}mat.uned.es

Hugo Parlier

IGAT Institute, EPFL
Bâtiment BCH
CH-1015 Lausanne
Switzerland
hugo.parlier{at}epfl.ch

We give a geometric characterization of compact Riemann surfaces admitting orientation–reversing involutions with fixed points. Such surfaces are generally called real surfaces and can be represented by real algebraic curves with non-empty real part. We show that there is a family of disjoint simple closed geodesics that intersect all geodesics of a pants decomposition at least twice in uniquely right angles if and only if such an involution exists. This implies that a surface is real if and only if there is a pants decomposition of the surface with all Fenchel–Nielsen twist parameters equal to 0 or 1/2.

The first author was supported in part by MTM 2005-01637 and the second author was supported in part by SNFS grant number PBEL2-106180.

2000 Mathematics Subject Classification 30F10, 32G15 (primary), 14H50, 30F20 (secondary).

Received February 28, 2006; revised April 27, 2007;
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