Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-25T13:34:46.559Z Has data issue: false hasContentIssue false
  • English
  • Français

Subgroups and subrings of profinite rings

Published online by Cambridge University Press:  24 October 2008

A. G. Abercrombie
A. G. Abercrombie
Affiliation:
Department of Pure Mathematics, University of Liverpool, P.O. Box 147, Liverpool L69 3BX
Get access Rights & Permissions [Opens in a new window]

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 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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 116 , Issue 2 , September 1994 , pp. 209 - 222
Copyright
Copyright © Cambridge Philosophical Society 1994

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Abercrombie, A. G.. The Hausdorff dimension of some exceptional sets of p-adic integer matrices. J. Number Theory (to appear).Google Scholar
[2]Billingsley, P.. Ergodic Theory and, Information (John Wiley, 1965).Google Scholar
[3]Christenson, C. O. and Voxman, W. L.. Aspects of Topology (Dekker, 1977).Google Scholar
[4]Cohn, D. L.. Measure Theory (Birkhäuser, 1980).Google Scholar
[5]Erdös, P. and Volkmann, B.. Additiver Gruppen mit vorgegebener Hausdorffscher Dimension. J. Reine. Angew. Math. 221 (1966), 203–208.Google Scholar
[6]Falconer, K.. The Geometry of Fractal Sets (Cambridge University Press, 1985).CrossRefGoogle Scholar
[7]Falconer, K.. Rings of fractional dimension. Mathematika 31 (1984), 25–27.CrossRefGoogle Scholar
[8]Falconer, K.. Hausdorff dimensions of distance sets. Mathematika 32 (1985), 206–212.CrossRefGoogle Scholar
[9]Hasse, H.. Number Theory Grundlehren 229 (Springer, 1980).Google Scholar
[10]Hewitt, E. and Ross, K.. Abstract Harmonic Analysis, vol. I (Springer, 1963).Google Scholar
[11]Hlawka, E.. The Theory of Uniform Distribution (AB Academic Publishers, 1984).Google Scholar
[12]Volkmann, B.. Eine metrische Eigenschaft reeller Zahlkörper. Math. Ann. 141 (1960), 237–238.CrossRefGoogle Scholar