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 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 > 10476.[1]
Application
Let Γd denote the Bianchi group PSL(2,Od), where Od is the
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
- ^ Zbl 0758.20009.
- Zbl 0254.10019.
- Maclachlan, Colin; Reid, Alan W. (2003). The Arithmetic of Hyperbolic 3-Manifolds. Zbl 1025.57001.