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,^[1] locally profinite groups,^[2] or t.d. groups^[3]). 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^[4] 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.^[2]

Tidy subgroups

Let G be a locally compact, totally disconnected group, U a compact open subgroup of G and ${\displaystyle \alpha }$ a continuous automorphism of G.

Define:

${\displaystyle U_{+}=\bigcap _{n\geq 0}\alpha ^{n}(U)}$

${\displaystyle U_{-}=\bigcap _{n\geq 0}\alpha ^{-n}(U)}$

${\displaystyle U_{++}=\bigcup _{n\geq 0}\alpha ^{n}(U_{+})}$

${\displaystyle U_{--}=\bigcup _{n\geq 0}\alpha ^{-n}(U_{-})}$

U is said to be tidy for ${\displaystyle \alpha }$ if and only if ${\displaystyle U=U_{+}U_{-}=U_{-}U_{+))$ and ${\displaystyle U_{++))$ and ${\displaystyle U_{--))$ are closed.

The scale function

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

Properties

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

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.