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. 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 on this subject was done in 1994, when George Willis showed that every locally compact totally disconnected group contains a so-called "tidy" subgroup and a special function on its automorphisms, the "scale function".

Tidy subgroups

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

Define:

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.

$s$ is continuous.
$s(x)=1$, whenever x in G is a compact element.
$s(x^n)=s(x)^n$ for every 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.

Sources

Source: G.A. Willis - [http://dz1.gdz-cms.de/no_cache/dms/load/img/?IDDOC=167209 The structure of totally disconnected, locally compact groups] , Mathematische Annalen 300, 341-363 (1994)