In this talk, we introduce a notion of sofic actions of groups on sets and graphs. While in works of Elek and Lippner and of Paunescu, notions of soficity in the context of pmp actions have already been developed, no satisfactory notions of sofic actions on discrete objects have heretofore been defined. This new notion of sofic actions allows us to strictly generalize the work of Hayes and Sale by showing a large class of generalized wreath product and graph wreath product groups are sofic. Interesting examples of sofic actions include transitive actions of sofic groups on graphs with amenable stabilizers, arbitrary actions of amenable groups and free groups on graphs, and arbitrary actions of LERF groups on sets. A majority of this talk is based on joint work with Srivatsav Kunnawalkam Elayavalli and Gregory Patchell.
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