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, or 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 by George Willis in 1994, opened up the field by showing that every locally compact totally disconnected group contains a so-called ''tidy'' subgroup and a special function on its automorphisms, the ''scale function'', giving a quantifiable parameter for the local structure. Advances on the ''global structure'' of totally disconnected groups were 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.

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 automorphisms 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 groups and linear groups over local skew fields by Helge Glöckner.

Notes

References

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*
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*{{citation
, last1=Willis , first1=G. , authorlink1=George A. Willis
, url=http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN002339951
, title=The structure of totally disconnected, locally compact groups
, journal= Mathematische Annalen
, volume=300
, pages=341-363
, date=1994
, doi=10.1007/BF01450491, url-access=subscription

Topological groups