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Prising apart geodesics by length in hyperbolic manifolds

Published online by Cambridge University Press:  24 March 2015

JAMES W. ANDERSON
JAMES W. ANDERSON*
Affiliation:
Mathematical Sciences, University of Southampton, Southampton, SO17 1BJ. email: [email protected]
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Abstract

We 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 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 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 ∂M have this property.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 158 , Issue 3 , May 2015 , pp. 547 - 572
Copyright
Copyright © Cambridge Philosophical Society 2015 

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