# 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[4] 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.[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*.

#### Properties

- is continuous.
- , whenever x in
*G*is a compact element. - for every non-negative 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

- Cartier 1979, §1.1
- Bushnell & Henniart 2006, §1.1
- Borel & Wallach 2000, Chapter X
- van Dantzig 1936, p. 411

## References

- van Dantzig, David (1936), "Zur topologischen Algebra. III. Brouwersche und Cantorsche Gruppen",
*Compositio Mathematica*,**3**: 408–426 - Borel, Armand; Wallach, Nolan (2000),
*Continuous cohomology, discrete subgroups, and representations of reductive groups*, Mathematical surveys and monographs,**67**(Second ed.), Providence, Rhode Island: American Mathematical Society, ISBN 978-0-8218-0851-1, MR 1721403 - Bushnell, Colin J.; Henniart, Guy (2006),
*The local Langlands conjecture for GL(2)*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences],**335**, Berlin, New York: Springer-Verlag, doi:10.1007/3-540-31511-X, ISBN 978-3-540-31486-8, MR 2234120 - Caprace, Pierre-Emmanuel; Monod, Nicolas (2011), "Decomposing locally compact groups into simple pieces",
*Math. Proc. Cambridge Philos. Soc.*,**150**: 97–128, arXiv:0811.4101, Bibcode:2011MPCPS.150...97C, doi:10.1017/S0305004110000368, MR 2739075 - Cartier, Pierre (1979), "Representations of -adic groups: a survey", in Borel, Armand; Casselman, William (eds.),
*Automorphic Forms, Representations, and L-Functions*(PDF), Proceedings of Symposia in Pure Mathematics, 33, Part 1, Providence, Rhode Island: American Mathematical Society, pp. 111–155, ISBN 978-0-8218-1435-2, MR 0546593 - G.A. Willis - The structure of totally disconnected, locally compact groups, Mathematische Annalen 300, 341-363 (1994)