Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

Subgroups and subrings of profinite rings

A. G. Abercrombiea1

a1 Department of Pure Mathematics, University of Liverpool, P.O. Box 147, Liverpool L69 3BX

Abstract

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 xs211D.

It is therefore natural to ask to what extent known results concerning Hausdorff dimension in xs211D 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 xs211D. 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 xs211D, where the analogous statement is known to be false.

(Received September 21 1993)

(Revised December 13 1993)