Trace equivalence in \(SU(2,1)\)
type: Item
instance of: scholarly article
title: Trace equivalence in \(SU(2,1)\)
zbMATH Open document ID: 0894.22006
DOI: 10.1023/A:1005079615846
author: Hanna Sandler
published in: Geometriae Dedicata
publication date: 1998-08-12T00:00:00Z
review text: 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 hav…
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 32Q45
Mathematics Subject Classification ID: 51M10
Mathematics Subject Classification ID: 53C10
Mathematics Subject Classification ID: 51M25
zbMATH DE Number: 1129449
zbMATH Keywords: trace equivalence
zbMATH Keywords: triangle groups
zbMATH Keywords: SU(2,1)
zbMATH Keywords: complex hyperbolic space
zbMATH Keywords: hyperbolic geometry
reviewed by: Charles L. Epstein
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1129449