Entropy of groups and subfactors

AbstractWe show that the classical notion of entropy of a finitely generated group G as introduced by A. Avez (C. R. Acad. Sci. Paris 275A, 1972, 1363–1366) is related by an explicit formula to the entropy of A. Connes and E. Størmer (Acta Math. 134, 1975, 288–306) and the index of V. F. R. Jones (Invent. Math. 72, 1983, 1–25) of the associated pair of finite von Neumann algebras as considered by S. Popa (C. R. Acad. Sci. Paris Sér. I Math. 309, 1989, 771–776). This construction is discussed in detail. We prove that the entropy of G is maximal if and only if G is the free group and compute its value. Then we show how the entropy of groups is connected to random walks on groups and information theory. Finally we give a brief discussion of the group entropy as a growth invariant and compare it to the Grigorchuk-Cohen cogrowth and Kesten's invariant λ(G)