2-generator arithmetic Kleinian groups

Abstract

We show that among the infinitely many conjugacy classes of finite covolume Kleinian groups generated by two elements of finite order, there are only finitely many which are arithmetic. In particular there are only finitely many arithmetic generalized triangle groups. This latter result generalizes a theorem of Takeuchi.

The (6,p)-arithmetic Hyperbolic Lattices in Dimension 3.

“…The cases where γ is real have been investigated in [30,31,35] We have shown in [33], that for each pair (p, q) there are only finitely many γ in C which yield arithmetic Kleinian groups and for all but a finite number of pairs (p, q), that finite number is zero. It is our aim here to determine all γ such that Γ is an arithmetic Kleinian group (i.e.…”

“…To eliminate these groups we now seek conditions on γ which force a discrete group Γ = f, g to be a free product. Moreover, we will extend the methods of [33] to enumerate the parameters γ which give rise to arithmetic Kleinian groups by obtaining bounds which involve the discriminant of the power basis of Q(γ) over L determined by γ. For this purpose, and also for other methods to be used in the enumeration, we want to obtain as stringent bounds as possible on |γ|, (γ), (γ).…”

“…The list obtained in Table 5 is produced by refining a basic list in §4.1 using arguments on the norm and discriminant, each stage being implemented by an elementary program in Maple. The finiteness of such a list was established in [33] and the starting point here uses the crude estimate obtained in [33] that p, q ≤ 120. In producing our lists, we assume that p, q ≥ 6 although the methods apply for p, q ≥ 3.…”

2-Generator arithmetic Kleinian groups III

“…Following [22] we define a Kleinian group G to be nearly arithmetic if G is a Kleinian subgroup of an arithmetic Kleinian group and G does not split as a nontrivial free product. Of course, an arithmetic Kleinian group is nearly arithmetic.…”

“…In previous work [22], [10] we established the finiteness of the number of two-generator arithmetic Kleinian groups generated by a pair of elliptic or parabolic elements. Further, we found there are exactly 4 arithmetic Kleinian groups generated by two parabolic elements [10], which are all knot and link complements.…”