Concept
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 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. 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 index of in is shown to be finite and independent of the U which is tidy for . Define the scale function as this index. Restriction to inner automorphisms gives a function on G with interesting properties. These are in particular: Define the function on G by where is the inner automorphism of on G. is continuous. whenever x in G is a compact element. for every non-negative integer . The modular function on G is given by .