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:
 * $$U_{+}=\bigcap_{n\ge 0}\alpha^n(U)$$
 * $$U_{-}=\bigcap_{n\ge 0}\alpha^{-n}(U)$$
 * $$U_{++}=\bigcup_{n\ge 0}\alpha^n(U_{+})$$
 * $$U_{--}=\bigcup_{n\ge 0}\alpha^{-n}(U_{-})$$

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.