Variations on a theme of Horowitz

Abstract

Horowitz showed that for every n at least 2, there exist elements w1, ..., wn in the free group F =free(a,b) which generate non-conjugate maximal cyclic subgroups of F and which have the property that trace(f(w1)) = ... =trace(f(wn)) for all faithful representations f of F into SL(2, C). Randol used this result to show that the length spectrum of a hyperbolic surface has unbounded multiplicity. Masters has recently extended this unboundness of the length spectrum to hyperbolic 3-manifolds. The purpose of this note is to present a survey of what is known about characters of faithful representations of F into SL(2, C), to give a conjectural topological characterization of such n-tuples of elements of F, and to discuss the case of faithful representations of general surface groups and 3-manifold groups.

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ORCID for J.W. Anderson: orcid.org/0000-0002-7849-144X

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