Arithmetic 2-generator Kleinian groups with quadratic invariant trace field
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.
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