"Geometry and Analysis on Groups" Research Seminar

Time: 2015.03.24, 15:00--17:00
Location: Seminarraum 8, Oskar-Morgenstern-Platz 1, 2.Stock
Title: "Ergodicity of algebraic group actions."
Speaker: Klaus Schmidt (Universität Wien)
Abstract: If $$G$$ is a countable discrete group, one can associate with every module $$M$$ over the integer group ring $$ZG$$ (via Pontryagin duality) an action of $$G$$ by automorphisms of the compact abelian group $$\hat M$$ dual to $$M$$. Such actions will be called 'algebraic $$G$$-actions' for short. Since such an algebraic $$G$$-action is determined by the module $$M$$, one should be able to characterise its dynamical properties in term of algebraic properties of $$M$$.

For $$G=Z^d$$, this connection is quite well understood, but if $$G$$ is nonabelian little is known about this connection at present.

The introductory part of this seminar will provide background on algebraic group actions, and the main part will focus on a simple class of modules (those of the form $$M=ZG/ZGf$$, where $$ZGf$$ is the principal left ideal generated by an element $$f\in ZG$$) and one of the simplest dynamical questions one can ask about such actions: when are they ergodic?

This is joint work with Hanfeng Li and Jesse Peterson.