Geometric intersection numbers on a four-punctured sphere

Abstract

Let G 4 \mathcal {G}_{4} be the space of all simple closed geodesics on the punctured sphere Σ 4 \Sigma _{4} . We construct an explicit homeomorphism of the completion of G 4 \mathcal {G}_{4} onto a circle by using geometric intersection numbers. Also, we relate these geometric intersection numbers to trace polynomials of transformations corresponding to geodesics in G 4 \mathcal {G}_{4} in a representation of π 1 ( Σ 4 ) \pi _{1}(\Sigma _{4}) into P S L ( 2 , C ) PSL(2,\mathbf {C}) .