We prove that the class of sofic groupoids is stable under several measure-theoretic constructions. In particular, we show that virtually sofic groupoids are sofic. We answer a question of Conley, Kechris, and Tucker-Drob by proving that an aperiodic pmp groupoid is sofic if and only if its full group is metrically sofic.
Nous démontrons dans cette note que plusieurs constructions de théorie de la mesure préservent la classe des groupoïdes sofiques. En particulier, nous montrons qu'un sous-groupoïde virtuellement sofique est sofique. Nous répondons aussi à une question de Conley, Kechris et Tucker-Drob en démontrant que, pour qu'un groupoïde apériodique muni d'une mesure de probabilité invariante soit sofique, il est nécessaire et suffisant que son groupe plein soit métriquement sofique.
Accepted:
Published online:
Luiz Cordeiro 1
@article{CRMATH_2018__356_9_957_0, author = {Luiz Cordeiro}, title = {On sofic groupoids and their full groups}, journal = {Comptes Rendus. Math\'ematique}, pages = {957--962}, publisher = {Elsevier}, volume = {356}, number = {9}, year = {2018}, doi = {10.1016/j.crma.2018.07.003}, language = {en}, }
Luiz Cordeiro. On sofic groupoids and their full groups. Comptes Rendus. Mathématique, Volume 356 (2018) no. 9, pp. 957-962. doi : 10.1016/j.crma.2018.07.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.07.003/
[1] Amenable Groupoids, Monogr. Enseign. Math., Monographs of L'Enseignement mathématique, vol. 36, L'Enseignement mathématique, Geneva, Switzerland, 2000 (With a foreword by Georges Skandalis and Appendix B by E. Germain)
[2] Entropy theory for sofic groupoids I: the foundations, J. Anal. Math., Volume 124 (2014), pp. 149-233
[3] Ultraproducts of measure preserving actions and graph combinatorics, Ergod. Theory Dyn. Syst., Volume 33 (2013) no. 2, pp. 334-374
[4] On groups of measure preserving transformations. II, Amer. J. Math., Volume 85 (1963), pp. 551-576
[5] Sofic dimension for discrete measured groupoids, Trans. Amer. Math. Soc., Volume 366 (2014) no. 2, pp. 707-748
[6] Full groups and soficity, Proc. Amer. Math. Soc., Volume 143 (2015) no. 5, pp. 1943-1950
[7] Sofic equivalence relations, J. Funct. Anal., Volume 258 (2010) no. 5, pp. 1692-1708
[8] Ergodic equivalence relations, cohomology, and von Neumann algebras. I, Trans. Amer. Math. Soc., Volume 234 (1977) no. 2, pp. 289-324
[9] Subrelations of ergodic equivalence relations, Ergod. Theory Dyn. Syst., Volume 9 (1989) no. 2, pp. 239-269
[10] Some extremely amenable groups related to operator algebras and ergodic theory, J. Inst. Math. Jussieu, Volume 6 (2007) no. 2, pp. 279-315
[11] Endomorphisms of symbolic algebraic varieties, J. Eur. Math. Soc., Volume 1 (1999) no. 2, pp. 109-197
[12] Locally compact groups and continuous logic, 2013 | arXiv
[13] Classical Descriptive Set Theory, Grad. Texts Math., vol. 156, Springer-Verlag, New York, 1995
[14] Global Aspects of Ergodic Group Actions, Math. Surv. Monogr., vol. 160, American Mathematical Society, Providence, RI, USA, 2010
[15] Invariants of orbit equivalence relations and Baumslag–Solitar groups, Tohoku Math. J. (2), Volume 66 (2014) no. 2, pp. 205-258
[16] Hyperlinearity, sofic groups and applications to group theory, 2009 http://www.kurims.kyoto-u.ac.jp/~narutaka/notes/NoteSofic.pdf (Unpublished notes)
[17] On sofic actions and equivalence relations, J. Funct. Anal., Volume 261 (2011) no. 9, pp. 2461-2485
[18] A convex structure on sofic embeddings, Ergod. Theory Dyn. Syst., Volume 34 (2014) no. 4, pp. 1343-1352
Cited by Sources:
Comments - Policy