Geometric moduli for Klein surfaces (Q757641)

This paper is concerned with the parametrisation of Teichmüller spaces for Klein surfaces. More precisely, let \(\Sigma\) be the interior of a compact topological surface \(\Sigma^*\) which is either non-orientable or has non-empty boundary or both; one then obtains an (open) Klein surface by endowing \(\Sigma\) with an analytic structure for which none of the boundary curves (if any) of \(\Sigma^*\) degenerates to a point, or equivalently with a complete hyperbolic metric d for which the boundary curves have infinite length. The problem is to find parameters which determine d up to an isometry homotopic to the identity mapping of \(\Sigma\). The authors show that, if \(\Sigma\) has genus \(g\geq 2\), then there exists 3g-3 closed curves on \(\Sigma\) such that the associated geodesic length functions do indeed determine d up to the required equivalence; in fact, they prove this when \(\Sigma\) is non-orientable with non-empty boundary, the other cases being already known. The result is interesting because (i) the Teichmüller space T(\(\Sigma\)) (which can be identified with a component of the Teichmüller space of real algebraic curves of genus g) has dimension exactly 3g-3, (ii) in the compact oriented case, more than 3g-3 curves are required to determine d.