Covolumes of lattices in simple Lie groups
A talk in the Mathematisches Kolloquium (TRR 358) series by
Ralf Köhl from CAU Kiel
| Abstract: | I will start my talk with the classical geometric group theory of the modular group, leading to the famous fundamental domain in the hyperbolic plane, the structure of the modular group as an amalgamated free product of finite cyclic groups, and the identification of the modular group as an oriented $(2,3,\infty)$ triangle group.
Studying Schwarz triangles (from 1873), one observes that the volume of the fundamental domain of the (oriented) $(2,3,7)$ triangle group is smaller than the fundamental domain of the modular group. Siegel (in 1945) established that the oriented $(2,3,7)$ triangle group indeed realizes a fundamental domain in the hyperbolic plane of minimal possible volume. Using the structure of the hyperbolic plane as a homogeneous space $SL(2,\mathbb R)/SO(2)$, this observation immediately leads to the fact that the oriented $(2,3,7)$ triangle group is a lattice of minimal covolume in the simple Lie group $SL(2,\mathbb R)$ with respect to the/any Haar measure. A general result by Kazhdan-Margulis (1968, while confirming Selberg's conjecture) shows that in fact any simple Lie group admits lattices of minimal covolume. This leads to the question which lattices these are. In my talk I will report on the current state of the art, recent progress by Thilmany (published 2019) and on some current work joint with Amir Dzambic and Kristian Holm. |