Prising apart geodesics by length in hyperbolic manifolds

Abstract

AbstractWe develop a condition on a closed curve on a surface or in a 3-manifold that implies that the length function associated to the curve on the space of all hyperbolic structures on the surface or in the 3-manifold (respectively) completely determines the curve. Specifically, for an orientable surfaceSof negative Euler characteristic, we extend the known result that simple curves have this property to curves with self-intersection number one (with one exceptional case arising from hyperellipticity that we describe completely). For a large class of hyperbolizable 3-manifolds, we show that curves freely homotopic to simple curves on ∂Mhave this property.