Homogeneous Ultrametric Spaces

Abstract

This is a continuation of our papers ‘‘Generalized Ultrametric Spaces,’’ Ž w x. I, II see 9, 10 , where we studied spaces X, endowed with an ultrametric distance d with values in a partially ordered set G. In those papers, we developed a full theory, which concerned equivalence relations compatible with the distance, the skeleton, the associated spherically complete space, and the Hahn space. A main topic was the consideration of immediate and of dense extensions as well as the question of embedding into the Hahn space. It should be pointed out that the ultrametric space X was not assumed to carry any further structure. The present paper is devoted to homogeneous ultrametric spaces}the main class of such spaces is obtained when a group G is acting transitively on the ultrametric space X. Accordingly, the results in the preceding papers give rise, in the situation of homogeneous spaces, to more specific statements, which reflect the homogeneity.