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In [[mathematics]], a '''totally disconnected group''' is a [[topological group]] that is [[totally disconnected]]. Such topological groups are necessarily [[Hausdorff space|Hausdorff]].

Interest centres on [[locally compact]] totally disconnected groups (variously referred to as groups of '''td-type''',<ref>{{harvnb|Cartier|1979|loc=§1.1}}</ref> [[locally profinite group]]s,<ref name=BushnellHenniart>{{harvnb|Bushnell|Henniart|2006|loc=§1.1}}</ref> '''t.d. groups'''<ref>{{harvnb|Borel|Wallach|2000|loc=Chapter X}}</ref>). The [[compact space|compact]] case has been heavily studied – these are the [[profinite group]]s – but for a long time not much was known about the general case. A theorem of [[David van Dantzig|van Dantzig]] from the 1930s, stating that every such group contains a compact [[open set|open]] [[subgroup]], was all that was known. Then groundbreaking work on this subject was done in 1994, when [[George Willis (mathematician)|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 group]]s and of Noetherian groups.

==Locally compact case==
{{main|Locally profinite group}}

In a locally compact, totally disconnected group, every [[neighbourhood (mathematics)|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.<ref name=BushnellHenniart/>

===Tidy subgroups===
Let ''G'' be a locally compact, totally disconnected group, ''U'' a compact open subgroup of ''G'' and <math>\alpha</math> a continuous automorphism of ''G''.

Define:

:<math>U_{+}=\bigcap_{n\ge 0}\alpha^n(U)</math>
:<math>U_{-}=\bigcap_{n\ge 0}\alpha^{-n}(U)</math>
:<math>U_{++}=\bigcup_{n\ge 0}\alpha^n(U_{+})</math>
:<math>U_{--}=\bigcup_{n\ge 0}\alpha^{-n}(U_{-})</math>

''U'' is said to be '''tidy''' for <math>\alpha</math> if and only if <math>U=U_{+}U_{-}=U_{-}U_{+}</math> and <math>U_{++}</math> and <math>U_{--}</math> are closed.

===The scale function===
The index of <math>\alpha(U_{+})</math> in <math>U_{+}</math> is shown to be finite and independent of the ''U'' which is tidy for <math>\alpha</math>. Define the scale function <math>s(\alpha)</math> as this index. Restriction to [[inner automorphism]]s gives a function on ''G'' with interesting properties. These are in particular: 
Define the function <math>s</math> on ''G'' by
<math>s(x):=s(\alpha_{x})</math>,
where <math>\alpha_{x}</math> is the inner automorphism of <math>x</math> on ''G''.

<math>s</math> is continuous. 
<math>s(x)=1</math>, whenever x in ''G'' is a compact element. 
<math>s(x^n)=s(x)^n</math> for every integer <math>n</math>. 
The modular function on ''G'' is given by <math>\Delta(x)=s(x)s(x^{-1})^{-1}</math>. 

===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 group]]s and linear groups over local skew fields by Helge Glöckner.

==Notes==
{{reflist}}

==References==
*{{Citation
| last=Borel
| first=Armand
| author-link=Armand Borel
| last2=Wallach
| first2=Nolan
| author2-link=Nolan Wallach
| title=Continuous cohomology, discrete subgroups, and representations of reductive groups
| year=2000
| edition=Second
| publisher=[[American Mathematical Society]]
| location=Providence, Rhode Island
| series=Mathematical surveys and monographs
| volume=67
| isbn=978-0-8218-0851-1
| mr=1721403
}}
*{{Citation | last1=Bushnell | first1=Colin J. | last2=Henniart | first2=Guy | title=The local Langlands conjecture for GL(2) | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] | isbn=978-3-540-31486-8 | doi=10.1007/3-540-31511-X | mr=2234120 | year=2006 | volume=335}}
*{{Citation | last1=Caprace | first1=Pierre-Emmanuel | last2=Monod | first2=Nicolas | title=Decomposing locally compact groups into simple pieces | journal=Math. Proc. Cambridge Philos. Soc. | doi=10.1017/S0305004110000368 | mr=2739075 | year=2011 | volume=150 | pages=97–128}}
*{{Citation
| last=Cartier
| first=Pierre
| author-link=Pierre Cartier (mathematician)
| contribution=Representations of <math>\mathfrak{p}</math>-adic groups: a survey
| year=1979
| title=Automorphic Forms, Representations, and L-Functions
| editor1-last=Borel
| editor1-first=Armand
| editor1-link=Armand Borel
| editor2-last=Casselman
| editor2-first=William
| editor2-link=William Casselman (mathematician)
| url=http://www.ams.org/online_bks/pspum331/pspum331-ptI-7.pdf
| publisher=[[American Mathematical Society]]
| publication-place=Providence, Rhode Island
| series=Proceedings of Symposia in Pure Mathematics
| volume=33, Part 1
| pages=111–155
| isbn=978-0-8218-1435-2
| mr=0546593
}}
*G.A. Willis - [http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=167209 The structure of totally disconnected, locally compact groups], [[Mathematische Annalen]] 300, 341-363 (1994)

[[Category:Topological groups]]

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