Grant CNCS-UEFISCDI
Sofic Groups and Ergodic Theory

Project code: PN-II-RU-TE-2014-4-0669

Contract no. 239/01.10.2015 financed by the Romanian National Authority for Scientific Research, CNCS - UEFISCDI

Period: October 2015- September 2017


Members



Abstract

The main objective of my study is to advance the use of the universal sofic action in order to provide a systematic investigation of soficity. The recently discovered obstructions will be generalised and exploited in order to ultimately prove the existence of non-sofic groups. A better understanding of these obstructions will give new tools for showing the soficity of new wide classes of groups. My aim is also to detect connections between other forms of soficity, like weak and linear soficity. I plan to develop further the description of weakly sofic groups provided by Glebsky and Rivera to prove weak soficity of concrete challenging examples, like the Higman group. By studying the linear groups over a finite field endowed with the rank metric I will explore the conjecture that every linear sofic group is actually sofic.


Expected results

We expect to provide further examples where ultrapower techniques are used to describe or prove properties of sofic groups and other metric approximation properties. Many examples use the action of elements of the universal sofic group onto the Loeb space, i.e. ultraproduct of probability spaces. We shall study the structure of these two objects and the way they interact.


Scientific seminars organized:

Publications produced as a result of this research

  • Liviu Pãunescu, Unitaries in Ultraproduct of Matrices - accepted to Jornal of Operator Theory.
  • Matteo Cavaleri, Computability of Følner sets -accepted to International Journal of Algebra and Computation.
  • B.M. Baker, T. Giordano, R.B. Munteanu, Approximate transitivity of the ergodic action of the group of finite permutations of N on {0,1}^N.
  • Matteo Cavaleri, Følner functions and the generic Word Problem for finitely generated amenable groups.
  • M. Cavaleri, R.B. Munteanu, L. Pãunescu Two special subgroups of the universal sofic group.
  • G. Arzhantseva, L. Pãunescu Constraint metric approximations and equations in groups.

    Conference participations and research visites

  • Universita Sapienza di Roma, September 16 - September 28 2017, research visit and international conference - Liviu Paunescu
  • Ecole Polytechnique Federale de Lausanne, May 3- May 12 2017, research visit and collaboration - Liviu Paunescu
  • University of Vienna, November 10- November 23 2016, research visit and collaboration - Liviu Paunescu
  • Hebrew University of Jerusalem, November 5- November 11 2016, international conference "60 Faces to Groups" - Matteo Cavaleri
  • Erwin Schrödinger Institute, January 17- March 31 2016, special semester "Measured Group Theory", research visit and collaboration - Liviu Paunescu
  • University of Vienna, October 25- November 7 2015, research visit and collaboration - Liviu Paunescu


    Reports
    Scientific report.