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Summary: CUSPS OF MINIMAL NON-COMPACT ARITHMETIC HYPERBOLIC
3-ORBIFOLDS
TED CHINBURG, DARREN LONG, AND ALAN W. REID
Abstract. In this paper we count the number of cusps of minimal non-compact finite volume
arithmetic hyperbolic 3-orbifolds. 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 non-compact finite volume arithmetic
hyperbolic 3-orbifolds. 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
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