Sifted character sums and free quotients of Bianchi groups

TL;DR

This paper demonstrates that Bianchi groups have large free quotients, with rank growing at least as fast as |d|^{1/4 - ε} as the discriminant |d| increases, revealing new structural properties of these groups.

Contribution

It establishes a lower bound on the rank of free quotients of Bianchi groups, a novel result in understanding their algebraic and geometric structure.

Findings

Bianchi groups have free quotients of rank at least |d|^{1/4 - ε}.

The rank of free quotients grows with the discriminant |d|.

The result applies as |d| tends to infinity.

Abstract

We show that the Bianchi group $\mathrm{PSL}_2(\mathcal{O}_d)$, where $\mathcal{O}_d$ is the ring of integers in $\Q(\sqrt{d})$, $d<0$, has a free quotient of rank $\geq |d|^{1/4-\epsilon}$, as $|d|\to\infty$.