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Subgroups and subrings of profinite rings

  • A. G. Abercrombie (a1)

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.

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[4] D. L. Cohn . Measure Theory (Birkhäuser, 1980).

[9] H. Hasse . Number Theory Grundlehren 229 (Springer, 1980).

[10] E. Hewitt and K. Ross . Abstract Harmonic Analysis, vol. I (Springer, 1963).

[12] B. Volkmann . Eine metrische Eigenschaft reeller Zahlkörper. Math. Ann. 141 (1960), 237238.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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