SU.SHING CHEN School of Mathematics Georgia Institute of Technology Atlanta, Georgia 30332 U.S.A. (Received April I0, 1978 in revised form October 3, 1978) ABSTRACT. A sufficient condition is given for a countable discrete group G to contain a free subgroup of two generators. KEY WORDS PHRASES. Nonamenable group, free group. 22D05. AMS (MOS) SUBJECT CLASSIFICATION (1970) CODES. Given a topological group G, we denote by L the Banach algebra of all real valued bounded left uniformly continuous functions on G with the supremem norm. A mean m on L is a continuous, positive, linear functional such that m(1) A mean is called invariant if m(f g) is the translate of f by g. I. re(f) for every f L g G, where fg G is called amenable if there exists an invarlant mean on L. G has the fixed point property if whenever G acts on a compact convex set Q afflnely in a locally convex topological vector space E, then G has a fixed point in Q [2]. S. CHEN It is well known that G is amenable if only if G has the fixed point property for any topological group G. In [4], von Neumann proved that if G has a free subgroup of two gene.rators then G is not amenable conjectured that the converse is true. In this paper, we shall give a sufficient condition for a discrete group G to contain a free subgroup of two generators. of von This result may" be interesting to the investigation Neumannâs conjecture. Let be an affine transformation of a compact convex set Q in a locally Then # has a fixed point in Q by the famous convex topological vector space E. Tychonoff fixed point theorem. point set Furthermore, one can prove easily that the fixed of Q is a compact convex subset of Q. The fixed point F# of an affine transformation Let us consider a discrete group G acting affinely on Q. set verse # F -1of each element of G coincides with the fixed point set F-I of the in- An element # of G is said to be attractive if for each weak neighbor- hood of the fixed point set F of the orbit {n(s)In of .} of any compact convex subset S in Q-U converges to the fixed point set is a positive integer N such that for all [n F > N,n(s) C U. that is, there of G An element is said to be weakly attractive if, for each weak neighborhood point set . (i) nNâ THEOREM. F# of of the fixed all there is a positive integer Nâ such that for (S) U. It is obvious that an attractive element weakly attractive. [Note: (i) , 7z. of G is {0}] If a discrete group G acts on a compact convex set of Q of a locally convex topological vector space E affinely such that G contains at least two weakly attractive elements without common fixed points, then G contains a free subgroup of two generators. PROOF. point sets Let be two weakly attractive elements of G. Then the fixed there exist F F# are disjoint. By the seperation theorem [6], NONAMENABLE GROUPS a linear functional L on E real numbers c for every x in that c c such that L x < c < c < Ly F every y F. in Without loss of generality, we may assume < 0 < c Thus K {x > 0} QILx < 0} is a weak convex neighborhood of K 2 {x QILx of K is a weak convex neighborhood of F. The complements K K respectively are compact convex sets in Q 2 1 Q- K By the definition of weak attractiveness, there exist positive integers Nâ N" such that N" Then t any relation section K nN (K) K1 nN ()C K2 id, we have for all n e* Let s Nâ while the group F generated by s t is a free group. In fact, forâ sPtq... {z e sPtq...(z) z for each z in the hyperplane Ic K2C QILz 0} of Q. But clearly LsPt q...(z) # 0, Lz We have a contradiction. If a nonamenable discrete group G acts on a compact convex set COROLLARY. Q of a locally convex topological vector space E affinely such that G contains all weakly attractive elements then G contains a free subgroup of two generators. PROOF. This follows from the theorem the non-fixed point property of nonamenable groups. ACKNOWLEDGEMENT. The author is indebted to the referee for his comments. S. CHEN
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