2-Generator arithmetic Kleinian groups III

Abstract

This paper forms part of the program to identify all the 2-generator arithmetic Kleinian groups. Here we identify all conjugacy classes of such groups with one generator parabolic and the other generator elliptic. There are exactly 14 of these and exactly 5 Bianchi groups in their commensurability class, namely PSL(2, O d ) for d = 1, 2, 3, 7 and 15. This complements our earlier identification of the 4 arithmetic Kleinian groups generated by two parabolic elements.

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

“…Also from §3, |γ 1 | ≤ 4. Furthermore, by (11), γ 2 = β 1 − γ 1 also corresponds to a group generated by an element of order 2 and an element of order p. Thus we can assume that the γ 1 -space is symmetric about (γ 1 ) = β 1 /2 and so…”

“…No values of c 0 (22,11,2) γ (22,11) No values of c 0 Here we consider the cases where r = 2 so that δ satisfies p(x) = x 2 +c 1 x+c 0 . From the Basic Methods we have obtained candidate values for c 1 and c 0 .…”

“…The four orientable hyperbolic 3manifolds with fundamental group generated by a pair of parabolic elements which are arithmetic are two bridge knot and link complements [21]. The 14 finite co-volume Kleinian groups with two generators, one of finite order, one parabolic, which are arithmetic are described in [11]. An algorithmic approach to deciding if an orbifold obtained by (p, 0) − (q, 0) surgery on a two bridge link (or knot) has arithmetic fundamental groups was given in [25].…”

Jørgensen number and arithmeticity

“…This is equivalent to Γ being commensurable with the Bianchi group PSL 2 (O d ), so it is useful to recall from Section 5 of [6] the presentations of several Bianchi groups (cf. [13] and [29]).…”

“…We also recall Theorem 1.1 from [6], which identifies all two-generator arithmetic Kleinian groups with one generator parabolic and the other elliptic. We further note the invariant trace field of each.…”