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 U which is tidy for $\alpha$. Define the scale function $s(\alpha )$ 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 $s(x):=s({\alpha }_x)$, where ${\alpha }_x$ is the inner automorphism of $x$ on G.

Properties

• $s$ is continuous.
• $s(x)=1$, 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 $\Delta (x)=s(x)s(x^{-1}{)}^{-1}$.

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.