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Length spectra as moduli for hyperbolic surfaces. (English) Zbl 0595.30052
Let S be a hyperbolic Riemann surface with a metric of constant negative curvature -1. It is known that in general even for compact S the geodesic length spectrum does not determine S up to isometry. By contrast, the main result of this paper is that if S is a torus with one puncture or one infinite area end (an h-torus) then the length spectrum does completely determine S.
When S has genus zero and three punctures or holes, a similar result follows easily from the facts that boundary geodesics have minimal length and that the lengths of the boundary geodesics determine the surface.
The key result in the case that S is an h-torus, of interest in its own right, is that a simple smooth closed geodesic on S minimises length in its homology class. Two geodesics on S are a generating pair for the fundamental group if and only if they are simple and have a single intersection. From results of Fricke and Keen, the lengths of a generating pair and of the boundary geodesic determine S. It is shown that the geodesic $$\alpha$$ of minimal length on S is simple, and that together with the next shortest simple geodesic $$\beta$$ it forms a generating pair. Using the result above about minimising length in the homology class, geodesics shorter than $$\beta$$ are characterized as belonging to Alt($$\alpha$$,$$\beta)$$, the subgroup of $$\pi_ 1(S)$$ generated by $$\alpha,\beta^{-1}\alpha \beta$$. (By abuse of notation $$\alpha$$ denotes both a geodesic and its representative in $$\pi_ 1.)$$ Finally, lengths of geodesics corresponding to elements of Alt($$\alpha$$,$$\beta)$$ are shown to be given by a two variable polynomial in the lengths of $$\alpha$$ and $$\alpha \beta \alpha^{-1}\beta^{-1}$$.
Reviewer: C.Series

##### MSC:
 30F20 Classification theory of Riemann surfaces 30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization) 20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
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