1 . 1 . 1997 on Problems Related to Growth , Entropy and Spectrum in Group

We review some known results and open problems related to the growth of groups. For a nitely generated group ?; given whenever necessary together with a nite generating set, we discuss the notions of (A) uniformly exponential growth, (B) growth tightness, (C) regularity of growth series, (D) classical growth series (with respect to word lengths), (E) growth series with respect to weights, (F) complete growth series, (G) spectral radius of simple random walks on Cayley graphs. From the modern point of view a dynamical system is a pair (G; X) where G is a group (or a semi-group) and X is a set on which G acts; introducing diierent structures on G and X we get diierent directions for the Theory of Dynamical Systems. Many of the dynamical properties of the pair (G; X) depend on appropriate properties of the group G: In this paper we discuss such notions for a group G as growth, entropy, amenability and spectral radius of random walks, which are closely related to the notions of growth of manifolds and foliations, entropy of geodesic ows and of symbolic systems, spectral theory of Laplace operators and the theory of harmonic functions on Riemannian manifolds. It is a great pleasure for the rst author to mention that one of the results that is discussed in this paper, namely the cogrowth criterium for amenability, was rstly reported at the seminar of Prof.