Proximity Relations in Transformation Groups

Introduction. Let (X,T) be a transformation group whose phase space X is a uniform space. Most of the results are stated for compact X. In this paper, four proximity relations, L, M, P, and Q, in X are defined and some of their properties studied. The relations P and Q were first defined and studied by Ellis and Gottschalk [5]. As a general reference for the notions occurring here, consult [8]. In Theorem 1, L is shown to be an invariant equivalence relation in X, and in Theorem 3, L is characterized as the union of all orbit closures under (X x X, T) which are contained in P. Theorem 3 also shows that if P is closed, then P = L and P is a closed invariant equivalence relation in X. Theorem 4 establishes the productivity of the relations Land M, and in Theorem 7, it is shown that (X,T) is coterminous, i.e., P = Q = L= M, iff P = Q. Theorems 9 through 13 describe L, M, P, and Q under various hypotheses such as (X,T) distal; (X,T) minimal and T abelian; (X, T) regionally mixing; and the like. Theorem 14 is an application of the general theory to obtain a characterization of (X,T) being uniformly equicontinuous, and as such represents a strengthening of a theorem by John D. Baum [2]. The author is indebted to Professor W. H. Gottschalk for his invaluable suggestions, mathematical stimulation, and sustained interest. STANDING NOTATION. Let (X,T) be a transformation group where X is always a uniform space. Let / be the class of all syndetic subsets of T, let XV be the class or all compact subsets of T, let OW be the uniformity of X, and for each x E X let

Higher order regionally proximal equivalence relations for general minimal group actions

Introduction. In this paper, we investigate the proximal relation and related notions in coset transformation groups.In §1, we show that many of the interesting properties of the proximal relation can be characterized in terms of a group subset that arises naturally from the generating subgroup.A general criterion for distal in terms of subnormality of the generating subgroup is shown in §2.Proofs are given for [9, Remark (2.1) and Theorem (2.2)], establishing that coset transformation groups of nilpotent groups are distal.§3 is concerned with distal in relation to certain coset transformation groups that arise from solvable Lie groups.Finally, we establish in §4 some results concerning the existence of subgroups which yield either point-transitive or minimal distal actions on the phase space.These results are strengthened when applied to certain discrete flows and examined in Lie groups.Proofs for the rest of the results of [9] are given in this section.Let G be a topological group and H a subgroup of G. Consider the space H\G={Hg | ge G} of right cosets of G.The coset transformation group of G induced by H is defined to be the point-to-point transitive transformation group (H\G, G) with action (Hfi g) -> Hfg.The fact that (H\G, G) is indeed a transformation group under this action follows from [6, Definition 1.39].If H is syndetic in G, H\G is compact uniformizable, whence H\G has a unique compatible uniformity.Thus, the proximal relation of (H\G, G) is well defined and will be denoted by P(H\G).Throughout this paper, G will denote a topological group with identity e and 77 will denote a syndetic subgroup.If K and L are subgroups of G, we denote that K is normal in L by K <] L. If K is a subgroup of G, then K* will denote the closure of K.The additive groups of integers and reals will be denoted by Z and R respectively.The neighborhood filter of e will be denoted by r¡e.For other definitions, see [9].Finally, the author would like to thank Professor W. Gottschalk for his stimulating comments concerning the results in this paper.2. The proximal relation.When the phase space is not necessarily uniform, we can still introduce a notion of proximal.

In this paper we study the relativized weak disjointness and the relativized regionally proximal relation for homomorphisms of point-transitive transformation groups, under the assumption of a relativized invariant measure. We also include a proof of a Folner-type result for syndetic subsets of an amenable group.