Gisele Teixeira Paula: Height estimates for Bianchi groups
Consider the geometry of the action of Bianchi groups \(\mathrm{SL}(2,\mathcal{O}_d)\) on the hyperbolic space \(\mathbb{H}^3\), where \(\mathcal{O}_d\) is the ring of integers of the imaginary quadratic field \(K = \mathbb{Q}(\sqrt{-d})\). We obtain, for some values of \(d\), an upper estimate for the height of some matrix \(M\) that takes a given point \((z,t) \in \mathbb{H}^3\) into the fundamental domain of the Bianchi group. This generalizes a lemma of Habegger and Pila about the action of the modular group on \(\mathbb{H}^2\). We use coarse fundamental domains that look like the so-called Siegel sets to make computations easier.
