Totally disconnected group

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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. 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 on this subject was done in 1994, when George Willis showed that every locally compact totally disconnected group contains a so-called tidy subgroup and a special function on its automorphisms, the scale function.

Contents 1 Tidy subgroups 2 The scale function 3 Calculations and applications 4 Sources

[edit] Tidy subgroups

Let G be a locally compact, totally disconnected group, U be a compact open subgroup of G and α a continuous automorphism of G.

Define:

U is said to be tidy for α if and only if U = U ₊ U ₋ = U ₋ U ₊ and U _{+ +} and U _{− −} are closed.

[edit] The scale function

The index of α(U ₊ ) in U ₊ is shown to be finite and independent of the U which is tidy for α. Define the scale function s(α) 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(α_x), where α_x is the inner automorphism of x on G.

s is continuous.
s(x) = 1, whenever x in G is a compact element.
s(xⁿ) = s(x)ⁿ for every integer n
The modular function on G is given by Δ(x) = s(x)s(x ^{− 1}) ^{− 1}

[edit] 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.

[edit] Sources

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

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