Simple closed geodesics of equal length on a torus

Summary

Abstract

Starting with a classical conjecture of Frobenius on solutions of the Markoff cubic, we are led, via the work of Harvey Cohn, to explore the multiplicities of lengths of simple geodesics on surfaces. We indicate recent progress on this and related questions stemming from the work of Schmutz Schaller. As an illustration we compare the cases of multiplicities on euclidean and hyperbolic once-punctured tori; in the euclidean case basic number theory gives a complete understanding of the spectrum. We explain an elementary construction using iterated Dehn twists that gives useful information about the lengths of simple geodesics in the hyperbolic case. In particular it shows that the marked simple length spectrum satisfies a rigidity condition: knowing just the order in the marked simple length spectrum is enough to determine the surface up to isometry. These results are special cases of a more general result [MP].

Introduction

The length spectrum of a hyperbolic surface is defined as the set of lengths of closed geodesics counted with multiplicities, and has been studied extensively in its relationship with the Laplace operator of a surface. A natural subset of the length spectrum is the simple length spectrum: the set of lengths of simple closed geodesics counted with multiplicities. This set is more naturally related to Teichmüller space and the mapping class group.