Higher torsion in the Abelianization of the full Bianchi groups

TL;DR

This paper investigates torsion phenomena in the integral homology of full Bianchi groups, revealing new examples of high prime torsion in their abelianizations, which were previously unobserved.

Contribution

It provides the first known examples of high prime torsion in the abelianization of full Bianchi groups, extending computational evidence beyond known bounds.

Findings

Detected torsion at prime p=80737 in the homology of Bianchi groups.

Found examples of p-torsion at primes much larger than the group's element orders.

Extended computational scope to identify torsion at discriminant -1747.

Abstract

Consider the Bianchi groups, namely the SL_2 groups over rings of imaginary quadratic integers. In the literature, there has been so far no example of p-torsion in the integral homology of the full Bianchi groups, for p a prime greater than the order of elements of finite order in the Bianchi group, which is at most 6. However, extending the scope of the computations, we can observe examples of torsion in the integral homology of the quotient space, at prime numbers as high as for instance p = 80737 at the discriminant -1747.