Totally disconnected group explained

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

See main article: Locally profinite group.

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 _{+} = c ap _{n\ge} \alpha ^{n(U)} U _{-} = c ap _{n\ge} \alpha ^{-n} (U) U _{++} = c up _{n\ge} \alpha ^{n(U} _{+} ) U _{--} = c up _{n\ge} \alpha ^{-n} (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 _{x} ) , where \alpha _{x} 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 ^{-1} ) ^{-1} .

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.

References

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