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Linear sofic groups and algebras


Authors: Goulnara Arzhantseva and Liviu Păunescu
Journal: Trans. Amer. Math. Soc. 369 (2017), 2285-2310
MSC (2010): Primary 20E26, 20C07, 16N99, 03C20, 20F70
DOI: https://doi.org/10.1090/tran/6706
Published electronically: April 8, 2016
MathSciNet review: 3592512
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Abstract: We introduce and systematically study linear sofic groups and linear sofic algebras. This generalizes amenable and LEF groups and algebras. We prove that a group is linear sofic if and only if its group algebra is linear sofic. We show that linear soficity for groups is a priori weaker than soficity but stronger than weak soficity. We also provide an alternative proof of a result of Elek and Szabó which states that sofic groups satisfy Kaplansky's direct finiteness conjecture.


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Goulnara Arzhantseva
Affiliation: Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria
Email: goulnara.arzhantseva@univie.ac.at

Liviu Păunescu
Affiliation: Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria – and – Institute of Mathematics of the Romanian Academy (on leave), 21 Calea Grivitei Street, 010702 Bucharest, Romania
Email: liviu.paunescu@imar.ro

DOI: https://doi.org/10.1090/tran/6706
Keywords: Sofic groups, metric ultraproduct, linear groups, Kaplansky's direct finiteness conjecture.
Received by editor(s): May 9, 2014
Received by editor(s) in revised form: March 22, 2015
Published electronically: April 8, 2016
Additional Notes: The first author was supported in part by the ERC grant ANALYTIC no. 259527, and by the Swiss NSF, under Sinergia grant CRSI22-130435
The second author was supported by the Swiss NSF, under Sinergia grant CRSI22-130435.
Article copyright: © Copyright 2016 American Mathematical Society