Examples of p-Adic Transformation Groups

The purpose of this paper is to give full descriptions of the examples announced in [8]. Let n be an integer, n > 2, and p a prime. Let AP denote the p-adic group (see below). Our principal result is the following. Example. There exists a compact metric space XI of dimension n and an action of AP on XI such that dim XI/Ap = n + 2. Such examples are interesting in relation to the important conjecture: If G is a compact group which acts effectively on a (generalized) manifold X, then G is a Lie group. This conjecture is proved for dim X ? 2 in [3] with generalizations given in [1], [9] and [10]. If this conjecture is false, then [4] there must be an example in which G = AP, and in this case [12], [2] dim X/Ap = n + 2. In this sense, the examples discussed here represent first approximations to counter-examples to the conjecture. The approximation is poor inasmuch as the spaces X" contain "large" n-dimensional sets; e.g., the n-skeleton of a triangulated (n+ 2)-manifold. Nor do we know of examples of AP action in which the dimension goes up by 3. It is known [12], [2] that the cohomology dimension cannot go up by more than 3. The authors expect to have more to say on these subjects. The reader might find the example of Kolmogoroff3 [5] a good introduction to the present paper. Kolmogoroff constructs a light open map which raises dimension by one; although not noted in [5], this map is the projection of an action of A2. One can construct analogous examples for all primes p, which as in [5], are based on the non-dimensional-full-valued spaces of Pontrjagin [7]. (A description of the Kolmogoroff example will appear in [11] as an illustration of the "A functor" of ? 2.) The additional complication in the examples described below stems from the following: If the dimension of a compact metric space X is to be decreased by 1 upon the removal of a subset D, D can be chosen to be totally disconnected. In contrast, to achieve a decrease of 2, D must have large components. Thus the individual modifications used in [7] are