We prove, using notions and techniques of topological dynamics, that a nonamenable group contains a finitely-generated subgroup of exponential growth. We also show that a group which belongs to a certain class, defined by means of topological dynamical properties, always contains a free subgroup on two generators. Topological dynamics can be viewed as a theory of representations of groups as homeomorphism groups. In this paper we want to emphasize this point of view and to show that information

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... about the structure of a group can be obtained by studying such representations. One illustration of this method is the characterization obtained in [7] of amenable groups as groups which admit no nontrivial strongly proximal flow. (See definitions below.) Furthermore this method yields results which do not explicitly involve notions from topological dynamics, for example using the above characterization of amenable groups we prove the theorem that a nonamenable group has a finitely generated subgroup of exponential growth. (This last result also follows from the main theorem in [8, p. 150] and Lemma 5 in [9] .) The paper is arranged as follows. In § 1 we define the necessary notions from topological dynamics. In particular strongly proximal, totally proximal and extremely proximal flows are defined. In the second section we prove, using the notion of a strongly proximal flow, that a nonamenable group contains a finitely generated subgroup of exponential growth [9] . In the third section we show that an extremely proximal flow is both strongly proximal and totally proximal. We show that a group which has a nontrivial extremely proximal flow contains a free subgroup on two generators. Finally we show by means of examples that strong proximality does not imply total proximality nor does total proximality imply strong proximality. I wish to thank Professor H. Furstenberg for his help and advice. 1. Definitions. A flow is a pair (T, X) consisting of a locally compact topological group T and a compact Hausdorff space X, equipped with a continuous function T x X -» X, denoted it, x) -► tx, which satisfies the following conditions: Fot all s, t eT and for all x e X, ist)x = sitx) and for all x e X ex = x,