Amenable groups and groups with the fixed point property

Abstract

fixed point property. This is the name given by Furstenberg to groups which have a fixed point every time they act affinely on a compact convex set in a locally convex topological linear space. Day [3] has shown that amenability implies the fixed point property. For discrete groups he has shown the converse. Along with amenable groups, we shall study, in this paper, groups with the fixed point property. This paper is based on part of the author's Ph.D. dissertation at Yale University. The author wishes to express his thanks to his adviser, F. J. Hahn. NOTATION. Group will always mean topological group. For a group G, Go will denote the identity component. Likewise Ho will be the identity component of H, etc. Banach spaces, topological vector spaces, etc., will always be over the real field. For topology, we use the notation of Kelley [18], except that our spaces will always