Arithmetic 2-generator Kleinian groups with quadratic invariant trace field
type: Item
instance of: scholarly article
zbMATH Open document ID: 0976.20033
publication date: 2001-09-27T00:00:00Z
Mathematics Subject Classification ID: 20H10
Mathematics Subject Classification ID: 30F40
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 1587786
zbMATH Keywords: cocompact arithmetic Kleinian groups
zbMATH Keywords: primitive elliptic elements
MaRDI profile type: Publication
author: C. Maclachlan
author: Gaven J. Martin
author: J. McKenzie
title: Arithmetic 2-generator Kleinian groups with quadratic invariant trace field
published in: New Zealand Journal of Mathematics
review text: The authors classify all cocompact arithmetic Kleinian groups that are generated by two elements of finite order and have quadratic invariant trace field. Roughly speaking, the strategy is the following. Let \(G\) be one such group, generated by two primitive elliptic elements \(f\) and \(g\) of orders \(p\), \(q\). It can be assumed that \(p,q\geq 3\), and it is not hard to see that \(p,q\in\{3,4,6\}\). The group \(G=\langle f,g\rangle\) is conjugate in \(\text{PSL}(2,\mathbb{C})\) to the subgroup generated by the images of the matrices NEWLINE\[NEWLINEX=\left[\begin{matrix}\cos(\pi/p)&i\sin(\pi/p)\\ i\sin(\pi/p)&\cos(\pi/p)\end{matrix}\right],\quad Y=\left[\begin{matrix}\cos(\pi/q)&it(\sin(\pi/q))\\ it^{-1}(\sin(\pi/q))&\cos(\pi/q)\end{matrix}\right],NEWLINE\]NEWLINE where \(|t|<1\). Then \(G\) is determined, for fixed \(p\) and \(q\), by NEWLINE\[NEWLINE\gamma(f,g)=\text{tr}[f,g]-2=(t-1/t)^2\sin^2(\pi/p)\sin^2(\pi/q).NEWLINE\]NEWLINE The authors bound the region in the complex plane in which \(\gamma(f,g)\) must lie, and deduce that there are only finitely many candidates. To further refine the possibilities for \(\gamma(f,g)\) they implement a version of the Dirichlet subroutine in Jeff Week's program SNAPPEA.NEWLINENEWLINENEWLINEThe paper is clearly written, but to understand it in full detail it is convenient to read simultaneously the paper by the first two authors ``The noncompact arithmetic generalised triangle groups'' that is known to the reviewer as a preprint and will be published elsewhere.
reviewed by: Jose Manuel Gamboa
description: scientific article