Density of Triple Coincidences in the Simple Length Spectrum of One-holed Tori

Abstract

We prove that, in the Teichmüller space of one-holed tori with any fixed boundary length, including the cuspidal case, marked hyperbolic one-holed tori with three distinct simple closed geodesics of equal length form a dense, meager subset. This answers a suspicion of McShane and Parlier in the negative. Moreover, every open set contains such a torus with common length above any prescribed threshold. To prove these results, we construct three families of simple closed curves in distinct Farey directions, estimate their lengths uniformly under twisting, and use the nondegeneracy of a two-dimensional length-ratio map to turn asymptotic agreement into an exact triple coincidence.