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.
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.
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