Geometry and Analysis on Groups Research Seminar

"Geometry and Analysis on Groups" Research Seminar

Time: 2018.11.29, 11:00–12:00
Location: Besprechungsraum, Oskar-Morgenstern-Platz 1, 2.Stock
Title: "Height estimates for Bianchi groups."
Speaker: Gisele Teixeira Paula (Université de Lille)
Abstract: 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 \(d\), a height estimate \(H(M)\leq cD(z,t)^9\), for some matrix \(M\) that takes a given point \((z,t)\in\mathbb{H}^3\) into the fundamental domain of the Bianchi group. Here, \(c\) is a constant that does not depend on the point and \(D(z,t)\) is an explicit function of the coordinates of the initial point. This generalizes a lemma of Habegger and Pila about the action of the modular group on \(\mathbb{H}^2\).