Zimmert set
# Zimmert set

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 $4n^2$ < -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 > $10^{476}.$

## Application

Let $Γ_d$ denote the Bianchi group $PSL(2,O_d),$ where $O_d$ is the ring of integers of. As a subgroup of PSL(2,C), there is an action of $Γ_d$ on hyperbolic 3-space $H_3,$ with a fundamental domain. It is a theorem that there are only finitely many values of d for which $Γ_d$ can contain an arithmetic subgroup G for which the quotient $H_3/G$ 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 $Γ_d$ and so the result above implies that almost all Bianchi groups have non-cyclic free quotients.

