The \((6,p)\)-arithmetic hyperbolic lattices in dimension 3. (Q652902)

It is known that there are infinitely many lattices in \(\mathrm{Isom}^+\mathbb H^3 \cong\mathrm{PSL}(2,\mathbb C)\) (equivalently Kleinian groups of finite covolume) which can be generated by two elements of finite orders \(p\) and \(q\). On the other hand the authors have shown previously that there are only finitely many arithmetic such lattices. This paper is part of a series devoted to identifying all these arithmetic \((p,q)\)-lattices. Here they deal with the \((6,p)\)-lattices. They prove that there are precisely \(53\) conjugacy classes of arithmetic \((6,p)\)-lattices and \(56\) Nielsen equivalent classes of pairs of generators. In every case \(p=2\), \(3\), \(4\) or \(6\). The arithmeticity of a lattice is determined by algebraic integers arising from the traces of preimages in \(\mathrm{SL}(2,\mathbb C)\) of a pair of generators of a \((6,p)\)-lattice.