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.

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.