Trace equivalence in \(SU(2,1)\) (Q1381336)

A problem considered originally by Fricke and later by Magnus is the characterization of those pairs of elements in a free group of rank \(n,\) \(F_n\), which have the same trace under all representations of \(F_n\) in \(Sl(2,R)\) where \(R\) is an integral domain of characteristic zero. Such elements are called \(Sl(2,R)\)-trace equivalent. For the cases \(R={\mathbb R, \mathbb C}\) and discrete representations, the trace has simple geometric interpretations. In particular, for loxodromic elements the length of the closed geodesic corresponding to the element is a simple function of its trace. Thus if two inconjugate elements always have the same trace then the closed geodesics they represent always have equal lengths. For the free group on two letters, \(F_2\), it is known that there are arbitrarily large families of inconjugate elements which are \(Sl(2,R)\)-trace equivalent. The paper under review considers the analogous question for representations of free groups in \(SU(2,1).\) Geometrically the analogy is between 2 (or 3) dimensional real hyperbolic geometry and 2-dimensional complex hyperbolic geometry. As the author points out trace equivalence in \(SU(2,1)\) implies trace equivalence in \(Sl(2,{\mathbb R})\), so the former property is a stronger condition. The author states that she knows of no pair of inconjugate elements in \(SU(2,1)\) which are \(SU(2,1)\)-trace equivalent. The main results of the paper are: 1. Restricting attention to the abstract group \(T={\mathbb Z}_2\star{\mathbb Z}_2\star{\mathbb Z}_2\) and a certain one parameter family of discrete representations \(T_\alpha\), \(\alpha\in[0,2\pi)\), the author proves: Theorem 1. For every positive integer \(N\) there exist inconjugate elements \(t_1,\dots, t_N\in T\) whose traces, under all the representations \(T_\alpha\), are equal. 2. The author constructs a map from \(F_2\) to the rational functions of one variable \(h\to R_h(x)\) such that: Theorem 3. Suppose that \(h, h'\in F_2\) which have the same trace under all representations of \(F_2\) in \(SU(2,1)\) then \(R_h=R_{h'}.\) Using this result, pairs of elements which are \(Sl(2,{\mathbb C})\)-trace equivalent but \(SU(2,1)\)-trace inequivalent are constructed.