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,[5] 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.[6]

Locally compact case

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.[2]

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 U_{+} in \alpha(U_{+}) is shown to be finite and independent of the compact open subgroup U which is tidy for \alpha. Defining the scale s(\alpha) of the continuous automorphism \alpha to be this index, we obtain the scale function s:\mathrm{Aut}(G)\to\mathbb{N}. Restricting s to inner automorphisms by setting s(x):=s(\alpha_{x}) for a given element x\in G with associated inner automorphism \alpha_{x} results in a function s:G\to\mathbb{N} with the following interesting properties.

Properties

  • s is continuous for the discrete topology on \mathbb{N}.
  • 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.

Notes

  1. ^ Cartier 1979, §1.1
  2. ^ Bushnell & Henniart 2006, §1.1
  3. ^ Borel & Wallach 2000, Chapter X
  4. ^ van Dantzig 1936, p. 411
  5. ^ Willis 1994
  6. ^ Caprace & Monod 2011

References