A Characterization of Arithmetic Subgroups ofSL (2, ℝ) andSL (2, ℂ)

Abstract

A Kleinian group is a discrete subgroup of PSL(2,C). As such it acts on 3-dimensional hyperbolic space H3. A Kleinian group G is said to have finite covolume if H3/G has finite volume. An interesting subclass of Kleinian groups of finite covolume is that of arithmetic Kleinian groups. Such a group is constructed as follows. Let k be a finite extension of Q with exactly one pair of complex conjugate embeddings (it may have many real embeddings), A a quaternion algebra over k which is ramified at all the real embeddings of k and O an order of A. We denote by O1 the elements of O of reduced norm 1. Via the complex embedding we have a map ρ:A↪M(2,C) and, in particular, O1↪SL(2,C). It can be shown that ρ(O1) is a discrete subgroup of SL(2,C), and projectivizing gives a discrete subgroup of PSL(2,C) which has finite covolume. An arithmetic Kleinian group is a subgroup of PSL(2,C) commensurable with some group Pρ(O1). These groups have many interesting properties and form a good set of examples with which to test conjectures in the theory of Kleinian groups and hyperbolic 3-manifolds. With this in mind it is an interesting problem to classify such groups in elementary terms. This was done by the reviewer in his thesis [see also C. Maclachlan and A. W. Reid, Math. Proc. Cambridge Philos. Soc. 102 (1987), no. 2, 251–257;], following the ideas of K. Takeuchi for characterizing arithmetic subgroups of PSL(2,R) [see J. Math. Soc. Japan 27 (1975), no. 4, 600–612;]. The main result of the paper under review is to rework these characterizations. As an application, the authors detail which orbifolds arising as cyclic branch covers of the Whitehead link are arithmetic.