Technische Wissenschaften, Physik, Mathematik
A systematic study of algebraic properties of random groups
Institut für Mathematik, Universität Wien
01.01.2014 - 31.12.2015
Kaplansky Zero Divisor Conjecture, Random Groups, Small Cancellation Theory, Hyperbolic Groups, Unique Product Groups

The present project aims to solve open problems in algebra. The research topic we focus on concerns infinite groups. The methods combine algebraic, geometric, and probabilistic tools.

Our main objective is to establish the famous Kaplansky zero-divisor conjecture for random finitely presented groups, that is - in a certain statistical sense - for almost all finitely presented groups. This outstanding conjecture states that the group ring of every torsion-free group over an integral domain has no zero divisors.

Kaplansky’s conjecture has been shown for various classes of groups, for example, unique product groups. On the other hand, Rips and Segev constructed infinitely many torsion-free hyperbolic groups without unique product property. Delzant showed that Kaplansky’s conjecture holds for any hyperbolic group with a certain large translation length property. Random finitely presented groups are torsion-free and hyperbolic. Thus, we plan to show that random finitely presented groups are unique product groups.

As a result of our study, we will obtain first results on the Kaplansky conjecture in the context of random groups. Thus, we plan to show, that Kaplansky’s conjecture holds for a typical finitely presented random group. Otherwise, we will determine a class of finitely presented groups which are potential counterexamples. Our investigation also contributes to the study of purely algebraic properties of hyperbolic groups. Moreover, our work will contribute in providing potential counterexamples to other outstanding conjectures.