Zimmert set

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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[edit]

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 4n2 < -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[edit]

For all but a finite number of d we have z(d) > 1: indeed this is true for all d > 10476.[1]

Application[edit]

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[2] and so the result above implies that almost all Bianchi groups have non-cyclic free quotients.[1]

References[edit]

  1. ^ a b Mason, A.W.; Odoni, R.W.K.; Stothers, W.W. (1992). "Almost all Bianchi groups have free, non-cyclic quotients". Math. Proc. Camb. Philos. Soc. 111 (1): 1–6. doi:10.1017/S0305004100075101. Zbl 0758.20009. 
  2. ^ Zimmert, R. (1973). "Zur SL2 der ganzen Zahlen eines imaginär-quadratischen Zahlkörpers". Inventiones mathematicae 19: 73–81. Zbl 0254.10019.