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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 In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ...
totally disconnected groups (variously referred to as groups of td-type, locally profinite groups, or t.d. groups). The
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Britis ...
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 In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgrou ...
, 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
automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphis ...
s, the ''scale function'', thereby advancing the knowledge of the local structure. Advances on the ''global structure'' of totally disconnected groups were obtained in 2011 by Caprace and
Monod Monod is a surname, and may refer to: * Adolphe Monod (1802–1856), French Protestant churchman; brother of Frédéric Monod. * Frédéric Monod (1794–1863), French Protestant pastor. * Gabriel Monod, French historian * Jacques Monod (1910� ...
, 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.


Tidy subgroups

Let ''G'' be a locally compact, totally disconnected group, ''U'' a compact open subgroup of ''G'' and \alpha a continuous automorphism of ''G''. Define: :U_=\bigcap_\alpha^n(U) :U_=\bigcap_\alpha^(U) :U_=\bigcup_\alpha^n(U_) :U_=\bigcup_\alpha^(U_) ''U'' is said to be tidy for \alpha if and only if U=U_U_=U_U_ and U_ and U_ are closed.


The scale function

The index of \alpha(U_) in U_ is shown to be finite and independent of the ''U'' which is tidy for \alpha. Define the scale function s(\alpha) as this index. Restriction to
inner automorphism In abstract algebra an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the ''conjugating element''. They can be realized via simple operations from within the group itse ...
s gives a function on ''G'' with interesting properties. These are in particular:
Define the function s on ''G'' by s(x):=s(\alpha_), where \alpha_ is the inner automorphism of x on ''G''.


Properties

* s is continuous. * s(x)=1, whenever x in ''G'' is a compact element. * s(x^n)=s(x)^n for every non-negative integer n. * The modular function on ''G'' is given by \Delta(x)=s(x)s(x^)^.


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 In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the ad ...
s and linear groups over local skew fields by Helge Glöckner.


Notes


References

* * * * *{{Citation , last=Cartier , first=Pierre , author-link=Pierre Cartier (mathematician) , contribution=Representations of \mathfrak{p}-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 The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
, 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
The structure of totally disconnected, locally compact groups
Mathematische Annalen 300, 341-363 (1994) Topological groups