Cusps of minimal non-compact arithmetic hyperbolic 3-orbifolds
The object of the paper under review is to count the number of cusps of minimal non-compact finite volume arithmetic hyperbolic 3-orbifolds. Recall that such an orbifold is isometric to \(\mathbb H^3/\Gamma\), where \(\mathbb H^3\) is the 3-dimensional hyperbolic upper half-space and \(\Gamma\) a maximal discrete arithmetic subgroup of \(\text{PGL}_2(k)\) for some imaginary quadratic number field \(k\). It is known that there is a unique maximal discrete arithmetic subgroup of \(\text{PGL}_2(k)\) which contains \(\text{PSL}_2({\mathfrak o}_k)\), where \({\mathfrak o}_k\) is the ring of integers of \(k\), and the number of cusps of this group was determined by \textit{E. B. Vinberg} [Sel. Math. Sov. 9, No. 4, 309--314 (1990; Zbl 0725.20037)]. The authors generalize Vinberg's result to arbitrary maximal arithmetic subgroups of \(\text{PSL}_2(k)\), and they conclude: Let \(C(N)\) be the set of isometry classes of minimal finite volume arithmetic hyperbolic three-orbifolds with exactly \(N\) cusps. Then only finitely many commensurability classes are represented by the elements of \(C(N)\). If \(C(N)\) is not empty, there are infinitely many elements of \(C(N)\) which are commensurable to each element of \(C(N)\). The proofs are mainly based on work of \textit{A. Borel} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 8, 1--33 (1981; Zbl 0473.57003); pp. 617--649 in: Collected Papers, Vol. 3 (1969--1982). Berlin etc.: Springer (1983; Zbl 0537.22001)], the Strong Approximation Theorem, and an argument of \textit{R. G. Swan} [Adv. Math. 6, 1--77 (1971; Zbl 0221.20060)] for constructing matrices satisfying various congruence conditions.
scientific article; zbMATH DE number 5380352