The Relation Between Wen s Theorem And Lawton - 1439 Words

In this work, we review a paper by Parker [4]. In [4] Parker derived several trace identities for M(3, C). In this paper we improve on the results obtained by Parker [4]. We discuss the relation between Wen’s theorem and Lawton [1] and Will’s work on the trace of the commutator [8]. We also present the merits on how to parametrise pair of pants via traces and cross-ratio. In the last section we use a formula which is due to Pratoussevitch [6] to compute trace of matrices generated by complex reflection. Mathematics Subject Classification : 51M99 Keywords: Hermitian matrix, Hermitian form, Free group, Triangle group, Special linear group, Special unitary group, Trace of a matrix. 1 Introduction In his survey paper published as [4], Parker studied the connection between the geometry of M and traces of Γ, where M is a complex hyperbolic orbifold written as H2 C /Γ and Γ is a discrete, faithful representation of π1(M) to Isom(H2 C). He did that by first considering the case where Γ is a free group on two generators and secondly, looking at the case where Γ is a triangle group generated by complex reflections in three complex lines. Several geometrical information connecting traces and complex hyperbolic space could be seen in Parker [4]. Pratossevitch [6] also presented several formulas for the traces of elements in complex hyperbolic triangle groups generated by complex reflections. In this paper we improve on the traces identities found by Parker [4] on M(3, C). One other main