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
http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.pnghttp://www.deepdyve.com/lp/hindawi-publishing-corporation/on-nonamenable-groups-MGPg7wyINF

“Hi guys, I cannot tell you how much I love this resource. Incredible. I really believe you've hit the nail on the head with this site in regards to solving the research-purchase issue.”

Daniel C.

“Whoa! It’s like Spotify but for academic articles.”

@Phil_Robichaud

“I must say, @deepdyve is a fabulous solution to the independent researcher's problem of #access to #information.”

@deepthiw

“My last article couldn't be possible without the platform @deepdyve that makes journal papers cheaper.”

Save this article to read later. You can see your Read Later on your DeepDyve homepage.

To save an article, log in first, or sign up for a DeepDyve account if you don’t already have one.

Subscribe to Journal Email Alerts

To subscribe to email alerts, please log in first, or sign up for a DeepDyve account if you don’t already have one.

Follow a Journal

To get new article updates from a journal on your personalized homepage, please log in first, or sign up for a DeepDyve account if you don’t already have one.