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 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 a continuous automorphism of G.
Define:

U is said to be tidy for if and only if and and are closed.

The scale function

The index of in is shown to be finite and independent of the compact open subgroup U which is tidy for. Defining the scale of the continuous automorphism to be this index, we obtain the scale function. Restricting to inner automorphisms by setting for a given element with associated inner automorphism results in a function with the following interesting properties.

Properties

is continuous for the discrete topology on.
, whenever x in G is a compact element.
for every non-negative integer.
The modular function on G is given by.

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.