The Arithmetic of Hyperbolic 3-Manifolds

Jørgensen number and arithmeticity

“…To determine which of these is arithmetic, we recall some basic facts about two-generator Kleinian groups and arithmeticity; these will also be useful in the next two sections. The first is a combination of (3.25) and Lemmas 3.5.7, 3.5.8, and 8.5.2 in [22]. • The trace field is Q(trΓ) = Q(trA, trB, trAB).…”

“…, which is a Z 2 -extension of the figure eight knot group by an involution that conjugates one generator to the other, but here Q(trG θ,k ) = Q(e πi/6 ) by Lemma 3.3, so this Z 2 -extension of the figure eight knot group is not conjugate to the one found in Case 1 since the trace field is a conjugacy invariant of finite-covolume Kleinian groups (see, for instance, Section 3.1 of [22]). If n is odd, then let…”

“…We next state Theorem 8.3.2 of [22], together with Lemma 4.1 in [9], which will enable us to determine when a finite-covolume Kleinian group is arithmetic.…”

“…For instance, all Jørgensen subgroups of the Picard group PSL 2 (O 1 ) are found in [11], and Theorem 3 of [16] This paper solves Problem 1.2 in two cases: torsion-free Kleinian groups and noncocompact arithmetic Kleinian groups (recall that a finite-covolume Kleinian group is non-cocompact if and only if it contains parabolic elements; see, for instance, Section 1.2 of [22]). …”

“…Identifying PSL 2 C with the orientation-preserving isometries of H 3 , we exploit in Section 2 the correspondence of torsion-free Kleinian groups Γ with orientable hyperbolic 3-manifolds H 3 /Γ (see, for instance, Section 1.3 of [22]) to prove the following result.…”