ENTROPY OF GROUPS AND SUBFACTORS

We 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). © 1992.