Totally disconnected group

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In mathematics, a totally disconnected group is a topological group that is totally disconnected. Such topological groups are necessarily Hausdorff. Interest centres on locally compact totally disconnected groups (variously referred to as groups of td-type,[1] locally profinite groups,[2] or t.d. groups[3]). The compact case has been heavily studied – these are the profinite groups – but for a long time not much was known about the general case. A theorem of van Dantzig[4] from the 1930s, stating that every such group contains a compact open subgroup, was all that was known. Then groundbreaking work by George Willis in 1994,[5] opened up the field by showing that every locally compact totally disconnected group contains a so-called tidy subgroup and a special function on its automorphisms, the scale function, giving a quantifiable parameter for the local structure. Advances on the global structure of totally disconnected groups were obtained in 2011 by Caprace and Monod, with notably a classification of characteristically simple groups and of Noetherian groups.[6]

Locally compact case

In a locally compact, totally disconnected group, every neighbourhood of the identity contains a compact open subgroup. Conversely, if a group is such that the identity has a neighbourhood basis consisting of compact open subgroups, then it is locally compact and totally disconnected.[2]

Tidy subgroups

Let G be a locally compact, totally disconnected group, U a compact open subgroup of G and α a continuous automorphism of G. Define:

U+=n0αn(U)
U=n0αn(U)
U++=n0αn(U+)
U=n0αn(U)

U is said to be tidy for α if and only if U=U+U=UU+ and U++ and U are closed.

The scale function

The index of α(U+) in U+ is shown to be finite and independent of the U which is tidy for α. Define the scale function s(α) as this index. Restriction to inner automorphisms gives a function on G with interesting properties. These are in particular:
Define the function s on G by s(x):=s(αx), where αx is the inner automorphism of x on G.

Properties

  • s is continuous.
  • s(x)=1, whenever x in G is a compact element.
  • s(xn)=s(x)n for every non-negative integer n.
  • The modular function on G is given by Δ(x)=s(x)s(x1)1.

Calculations and applications

The scale function was used to prove a conjecture by Hofmann and Mukherja and has been explicitly calculated for p-adic Lie groups and linear groups over local skew fields by Helge Glöckner.

Notes

References

  • van Dantzig, David (1936), "Zur topologischen Algebra. III. Brouwersche und Cantorsche Gruppen", Compositio Mathematica, 3: 408–426
  • Borel, Armand; Wallach, Nolan (2000), Continuous cohomology, discrete subgroups, and representations of reductive groups, Mathematical surveys and monographs, vol. 67 (Second ed.), Providence, Rhode Island: American Mathematical Society, ISBN 978-0-8218-0851-1, MR 1721403
  • Bushnell, Colin J.; Henniart, Guy (2006), The local Langlands conjecture for GL(2), Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 335, Berlin, New York: Springer-Verlag, doi:10.1007/3-540-31511-X, ISBN 978-3-540-31486-8, MR 2234120
  • Caprace, Pierre-Emmanuel; Monod, Nicolas (2011), "Decomposing locally compact groups into simple pieces", Mathematical Proceedings of the Cambridge Philosophical Society, 150: 97–128, arXiv:0811.4101, Bibcode:2011MPCPS.150...97C, doi:10.1017/S0305004110000368, MR 2739075
  • Cartier, Pierre (1979), "Representations of 𝔭-adic groups: a survey", in Borel, Armand; Casselman, William (eds.), Automorphic Forms, Representations, and L-Functions (PDF), Proceedings of Symposia in Pure Mathematics, vol. 33, Part 1, Providence, Rhode Island: American Mathematical Society, pp. 111–155, ISBN 978-0-8218-1435-2, MR 0546593
  • Willis, G. (1994), "The structure of totally disconnected, locally compact groups", Mathematische Annalen, 300: 341–363, doi:10.1007/BF01450491