Variations on a theme of Horowitz

Horowitz [Hor1] showed that for every n ≥ 2, there exist elements w1, . . . , wn in F2 = free(a, b) which generate non-conjugate maximal cyclic subgroups of F2 and which have the property that trace(ρ(w1)) = · · · = trace(ρ(wn)) for all faithful representations ρ of F2 into SL2(C). Randol [Ran] used this result to show that the length spectrum of a hyperbolic surface has unbounded multiplicity. Masters [Mas] 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 F2 into SL2(C), to give a conjectural topological characterization of such n-tuples of elements of F2, and to discuss the case of faithful representations of general surface groups and 3-manifold groups.