a1 Department of Pure Mathematics, University of Liverpool, P.O. Box 147, Liverpool L69 3BX
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 .
It is therefore natural to ask to what extent known results concerning Hausdorff dimension in carry over to the profinite setting. In this paper, following a line of thought initiated by B. Volkmann in , 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 . 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 , where the analogous statement is known to be false.
(Received September 21 1993)
(Revised December 13 1993)