SUBGROUPS AND SUBRINGS OF PROFINITE RINGS

A profinite topological group is compact and therefore possesses a unique invariant probability measure (Haar measure). We shall see that it is possible to define a fractional dimension on such a group in a canonical way, making use of Haar measure and a natural choice of invariant metric. This fractional dimension is analogous to Hausdorff dimension in R. It is therefore natural to ask to what extent known results concerning Hausdorff dimension in R carry over to the profinite setting. In this paper, following a line of thought initiated by B. Volkmann in [12], we consider rings of a-adic integers and investigate the possible dimensions of their subgroups and subrings. We will find that for each prime p the ring of p-adic integers possesses subgroups of arbitrary dimension. This should cause little surprise since a similar result is known to hold in R. However, we will also find that there exists a ring of a-adic integers possessing Borel subrings of arbitrary dimension. This is in contrast with the situation in R, where the analogous statement is known to be false.