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,[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

Main article: Locally profinite group

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 ${\displaystyle \alpha }$ a continuous automorphism of G.

Define:

${\displaystyle U_{+}=\bigcap _{n\geq 0}\alpha ^{n}(U)}$

${\displaystyle U_{-}=\bigcap _{n\geq 0}\alpha ^{-n}(U)}$

${\displaystyle U_{++}=\bigcup _{n\geq 0}\alpha ^{n}(U_{+})}$

${\displaystyle U_{--}=\bigcup _{n\geq 0}\alpha ^{-n}(U_{-})}$

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

The scale function

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

Properties

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

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

1. Cartier 1979, §1.1
2. Bushnell & Henniart 2006, §1.1
3. Borel & Wallach 2000, Chapter X
4. van Dantzig 1936, p. 411
5. Willis 1994
6. Caprace & Monod 2011

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 (1): 97–128, arXiv:0811.4101, Bibcode:2011MPCPS.150...97C, doi:10.1017/S0305004110000368, MR 2739075
• Cartier, Pierre (1979), "Representations of ${\displaystyle {\mathfrak {p}}}$-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