Prising apart geodesics by length in hyperbolic 3-manifolds

Details Prising apart geodesics by length in hyperbolic 3-manifolds Prising apart geodesics by length in hyperbolic 3-manifolds

2012-02-04

arxiv.org/abs/1202.0905v1

Mathematics

Geometric Topology

16 pages

Scientific paper

In this note, we develop a condition on a closed curve on a surface or in a 3-manifold that implies that the curve has the property that its length function on the space of all hyperbolic structures on the surface or 3-manifold completely determines the curve. For an orientable surface $S$ of 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 on closed surfaces of genus two that we describe completely), while for hyperbolizable 3-manifolds, we show that curves freely homotopic to simple curves on $\partial M$ have this property.

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