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 $$\mathbb{H}^2$$.