Totally disconnected group
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A totally disconnected, locally compact group is a topological group with these two properties.
For a long time not much was known about these groups. A theorem of van Dantzig - that every such group contains a compact, open subgroup - from the 1930's was all that was known.
Then groundbreaking work on this subject was done in 1994, when George Willis showed, that every totally disconnected, locally compact group contains a so called tidy subgroup and a special function on its automorphisms, the scale function.
Tidy subgroups
Let G be a locally compact, totally disconnected group, U be a compact open subgroup of G and [\alpha] a continuous automorphism of G.Define:
[U_=\bigcup_\alpha(U)]
[U_=\bigcup_\alpha^(U)]
[U_=\bigcup_\alpha(U_)]
[U_=\bigcup_\alpha^(U_)]
U is said to be tidy iff [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_)], where [\alpha_] is the inner automorphism of [x] on G.
[s] is continuous.
[s(x)=1], whenever x in G is a compact element.
[s(x^n)=s(x)^n] for every integer [n]
The modular function on G is given by [\Delta(x)=s(x)s(x^)^]
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.Sources
Source: G.A. Willis - The structure of totally disconnected, locally compact groups, Mathematisch Annalen 300, 341-363 (1994)
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