** "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 \(\mathbb{H}^2\).