On Finitely Generated Groups and Problems Related to Growth

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. Growth considerations, with motivations from diierential geometry, have been introduced in the early 50 's ((Sch], Efr], as well as Fol]) and again (independently !) in the late 60 's Milnor]. Nowadays, it is part of overlapping subjects with names such as combinatorial group theory, geometric group theory, or asymptotic group theory (the latter names being apparently rst used in Gk6]). These theories have a rich history ChM], for which milestones have been the 1966 book by Magnus, Karas and Solitar MKS], and the 1987 list of problems of R. Lyndon Lyn] (see also LyS]). For a recent state of the art concerning innnite groups, a quasi-isometric picture of the subject has appeared in Gv4]; for the asymptotic group theory of nite groups, see Kan]. For growth of other objects, see the following papers, as well as the references therein. The rst author acknowledges support from the \Fonds National Suisse de la Recherche Scientiique", from the \Russian Fund for Fundamental Research" (96-01-00974), and from INTAS (94-3420).