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In mathematics, a Zimmert set is a set of positive integers associated with the structure of quotients of hyperbolic three-space by a Bianchi group.
Definition
Fix an integer d and let D be the discriminant of the imaginary quadratic field Q(√-d). The Zimmert set Z(d) is the set of positive integers n such that 4n < -D-3 and n ≠ 2; D is a quadratic non-residue of all odd primes in d; n is odd if D is not congruent to 5 modulo 8. The cardinality of Z(d) may be denoted by z(d).
Property
For all but a finite number of d we have z(d) > 1: indeed this is true for all d > 10.
Application
Let Γd denote the Bianchi group PSL(2,Od), where Od is the ring of integers of. As a subgroup of PSL(2,C), there is an action of Γd on hyperbolic 3-space H3, with a fundamental domain. It is a theorem that there are only finitely many values of d for which Γd can contain an arithmetic subgroup G for which the quotient H3/G is a link complement. Zimmert sets are used to obtain results in this direction: z(d) is a lower bound for the rank of the largest free quotient of Γd and so the result above implies that almost all Bianchi groups have non-cyclic free quotients.
References
- ^ Mason, A.W.; Odoni, R.W.K.; Stothers, W.W. (1992). "Almost all Bianchi groups have free, non-cyclic quotients". Mathematical Proceedings of the Cambridge Philosophical Society. 111 (1): 1–6. Bibcode:1992MPCPS.111....1M. doi:10.1017/S0305004100075101. S2CID 122325132. Zbl 0758.20009.
- Zimmert, R. (1973). "Zur SL2 der ganzen Zahlen eines imaginär-quadratischen Zahlkörpers". Inventiones Mathematicae. 19: 73–81. Bibcode:1973InMat..19...73Z. doi:10.1007/BF01418852. S2CID 121281237. Zbl 0254.10019.
- Maclachlan, Colin; Reid, Alan W. (2003). The Arithmetic of Hyperbolic 3-Manifolds. Graduate Texts in Mathematics. Vol. 219. Springer-Verlag. ISBN 0-387-98386-4. Zbl 1025.57001.