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, 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 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", thereby advancing the knowledge of the local structure. Advances on the "global structure" of totally disconnected groups have been 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 formula_1 a continuous automorphism of "G".
Define:
"U" is said to be tidy for formula_1 if and only if formula_7 and formula_8 and formula_9 are closed.
The scale function.
The index of formula_10 in formula_11 is shown to be finite and independent of the "U" which is tidy for formula_1. Define the scale function formula_13 as this index. Restriction to inner automorphisms gives a function on "G" with interesting properties. These are in particular:<br>
Define the function formula_14 on "G" by 
formula_15, 
where formula_16 is the inner automorphism of formula_17 on "G".
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.