# 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**,^{[1]} locally profinite groups,^{[2]} **t.d. groups**^{[3]}). 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 clasification of characteristically simple groups and of Noetherian groups.

## Locally compact case

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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.^{[2]}

### 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 *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 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.

## Notes

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- ↑
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## References

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- G.A. Willis - The structure of totally disconnected, locally compact groups, Mathematische Annalen 300, 341-363 (1994)