Geometry and Analysis on Groups Research Seminar

"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.