Characters and lengths of geodesics in hyperbolic 3-manifolds

Abstract

According to the work by Randol, there exists pairs of closed curves on a surface S for which the geodesics in their respective homotopy classes have the same hyperbolic length, irrespective of the hyperbolic structure on S. In this work we look at this result in connection to hyperbolic 3-manifolds, and in particular the book of I-bundles manifold. We consider the following problem. Let G = 1fl(M), where M is a compact hyperbolizable 3-manifold, and consider all faithful representations of G into SL2(C). Find a topological condition P that can be imposed on the elements of G so the following is true. If 9 E G satisfies condition P, and h EGis any element such that X[h] = X[g] then h is conjugate to g±l. Here X[gJ is the character of g, which is defined in terms of the trace of the matrix representation of gin SL2(C). This problem can be translated into a question about the lengths of the geodesics in M by utilizing the connection between the character of an element of G, and the length of its geodesic representative in M. We therefore look for a property that gives some geometrical information about the manifold. For the purpose of this work the manifold M is a book of I-bundles.

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