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,[1] locally profinite groups,[2] or t.d.
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[4] 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,[5] 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.
[6] 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.
a continuous automorphism of G. Define: U is said to be tidy for
is shown to be finite and independent of the U which is tidy for
Restriction to inner automorphisms gives a function on G with interesting properties.
on G. 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.