Totally disconnected group

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, locally profinite groups, or t.d. groups). 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 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, 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.

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.

Tidy subgroups

Let G be a locally compact, totally disconnected group, U a compact open subgroup of G and [math]\displaystyle{ \alpha }[/math] a continuous automorphism of G.

Define:

[math]\displaystyle{ U_{+}=\bigcap_{n\ge 0}\alpha^n(U) }[/math]

[math]\displaystyle{ U_{-}=\bigcap_{n\ge 0}\alpha^{-n}(U) }[/math]

[math]\displaystyle{ U_{++}=\bigcup_{n\ge 0}\alpha^n(U_{+}) }[/math]

[math]\displaystyle{ U_{--}=\bigcup_{n\ge 0}\alpha^{-n}(U_{-}) }[/math]

U is said to be tidy for [math]\displaystyle{ \alpha }[/math] if and only if [math]\displaystyle{ U=U_{+}U_{-}=U_{-}U_{+} }[/math] and [math]\displaystyle{ U_{++} }[/math] and [math]\displaystyle{ U_{--} }[/math] are closed.

The scale function

The index of [math]\displaystyle{ \alpha(U_{+}) }[/math] in [math]\displaystyle{ U_{+} }[/math] is shown to be finite and independent of the U which is tidy for [math]\displaystyle{ \alpha }[/math]. Define the scale function [math]\displaystyle{ s(\alpha) }[/math] as this index. Restriction to inner automorphisms gives a function on G with interesting properties. These are in particular:
Define the function [math]\displaystyle{ s }[/math] on G by [math]\displaystyle{ s(x):=s(\alpha_{x}) }[/math], where [math]\displaystyle{ \alpha_{x} }[/math] is the inner automorphism of [math]\displaystyle{ x }[/math] on G.

Properties

• [math]\displaystyle{ s }[/math] is continuous.
• [math]\displaystyle{ s(x)=1 }[/math], whenever x in G is a compact element.
• [math]\displaystyle{ s(x^n)=s(x)^n }[/math] for every non-negative integer [math]\displaystyle{ n }[/math].
• The modular function on G is given by [math]\displaystyle{ \Delta(x)=s(x)s(x^{-1})^{-1} }[/math].

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.

References

• van Dantzig, David (1936), "Zur topologischen Algebra. III. Brouwersche und Cantorsche Gruppen", Compositio Mathematica 3: 408–426, http://www.numdam.org/item?id=CM_1936__3__408_0 
• Borel, Armand; Wallach, Nolan (2000), Continuous cohomology, discrete subgroups, and representations of reductive groups, Mathematical surveys and monographs, 67 (Second ed.), Providence, Rhode Island: American Mathematical Society, ISBN 978-0-8218-0851-1 
• Bushnell, Colin J.; Henniart, Guy (2006), The local Langlands conjecture for GL(2), Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 335, Berlin, New York: Springer-Verlag, doi:10.1007/3-540-31511-X, ISBN 978-3-540-31486-8 
• Caprace, Pierre-Emmanuel; Monod, Nicolas (2011), "Decomposing locally compact groups into simple pieces", Math. Proc. Cambridge Philos. Soc. 150: 97–128, doi:10.1017/S0305004110000368, Bibcode: 2011MPCPS.150...97C 
• Cartier, Pierre (1979), "Representations of [math]\displaystyle{ \mathfrak{p} }[/math]-adic groups: a survey", in Borel, Armand; Casselman, William, Automorphic Forms, Representations, and L-Functions, Proceedings of Symposia in Pure Mathematics, 33, Part 1, Providence, Rhode Island: American Mathematical Society, pp. 111–155, ISBN 978-0-8218-1435-2, http://www.ams.org/online_bks/pspum331/pspum331-ptI-7.pdf 
• G.A. Willis - The structure of totally disconnected, locally compact groups, Mathematische Annalen 300, 341-363 (1994)