 
Summary: CUSPS OF MINIMAL NONCOMPACT ARITHMETIC HYPERBOLIC
3ORBIFOLDS
TED CHINBURG, DARREN LONG, AND ALAN W. REID
Abstract. In this paper we count the number of cusps of minimal noncompact finite volume
arithmetic hyperbolic 3orbifolds. We show that for each N, the orbifolds of this kind which have
exactly N cusps lie in a finite set of commensurability classes, but either an empty or an infinite
number of isometry classes.
1. Introduction.
In this paper we count the number of cusps of minimal noncompact finite volume arithmetic
hyperbolic 3orbifolds. An orbifold of this kind is isometric to H3
/, where H3
is hyperbolic upper
half space and is a maximal discrete arithmetic subgroup in PGL2(k) for some imaginary quadratic
field k.
It is well known (cf. §3 below) that the cusps of the orbifold H3
/ correspond to equivalence
classes of points of P1
k under the action of PGL2(k) on P1
k. It was first noted by Bianchi [2] that
H3
