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

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.

Tidy subgroups

Let G be a locally compact, totally disconnected group, U a compact open subgroup of G and $\alpha$ a continuous automorphism of G.

Define:

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

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

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

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

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

The scale function

The index of $\alpha (U_{+})$ in $U_{+}$ is shown to be finite and independent of the compact open subgroup U which is tidy for $\alpha$. Defining the scale $s(\alpha )$ of the continuous automorphism $\alpha$ to be this index, we obtain the scale function $s:\mathrm{A}\mathrm{u}\mathrm{t}(G)\to \mathbb{\mathbb{N}}$. Restricting $s$ to inner automorphisms by setting $s(x):=s({\alpha }_x)$ for a given element $x\in G$ with associated inner automorphism ${\alpha }_x$ results in a function $s:G\to \mathbb{\mathbb{N}}$ with the following interesting properties.

Properties

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

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.

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", Mathematical Proceedings of the Cambridge Philosophical Society 150 (1): 97–128, doi:10.1017/S0305004110000368, Bibcode: 2011MPCPS.150...97C 
• Cartier, Pierre (1979), "Representations of $\mathfrak{𝔭}$-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 
• "The structure of totally disconnected, locally compact groups", Mathematische Annalen 300: 341-363, 1994, doi:10.1007/BF01450491, http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN002339951