Sofic entropy and amenable groups (Q2908140)

Sofic groups were introduced implicitly by \textit{M. Gromov} [J. Eur. Math. Soc. (JEMS) 1, No. 2, 109--197 (1999; Zbl 0998.14001)] and explicitly by \textit{B. Weiss} [Sankhyā, Ser. A 62, No. 3, 350--359 (2000; Zbl 1148.37302)]; both residually finite and amenable groups are sofic. The author [J. Am. Math. Soc. 23, No. 1, 217--245 (2010; Zbl 1201.37005)] introduced a new family of entropy-like invariants for measurable isomorphisms of actions of sofic groups. Here it is shown that this sofic entropy coincides with the classical entropy for actions of amenable groups. The proof involves the creation of a new invariant of measurable isomorphism, the upper-sofic entropy, and the deep work of \textit{D. J. Rudolph} and \textit{B. Weiss} [Ann. Math. (2) 151, No. 3, 1119--1150 (2000; Zbl 0957.37003)] developing entropy theory for orbit equivalences of amenable group actions relative to the orbit-change algebra.