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Published: 
January 7, 2004

Keywords: 
locally compact group, scale function, tidy subgroup, modular function, automorphism

Subject: 
Primary 22D05; Secondary 22D45, 20E25, 20E36

Abstract:

Let $\alpha$ be an automorphism of the totally disconnected group $G$.
The compact open subgroup, $V$, of $G$ is
\emph{tidy} for
$\alpha$ if $[\alpha(V') : \alpha(V')\cap V']$ is
minimised at $V$, where $V'$ ranges over all compact open subgroups
of $G$. Identifying a subgroup tidy for $\alpha$ is analogous to
identifying a basis which puts a linear transformation into Jordan
canonical form. This analogy is developed here by showing that commuting
automorphisms have a common tidy subgroup of $G$ and, conversely,
that a group $\siH$ of automorphisms having a common tidy
subgroup $V$ is abelian modulo the automorphisms which leave
$V$ invariant. Certain subgroups of $G$ are the analogues of
eigenspaces and corresponding real characters of $\siH$ the
analogues of eigenvalues.

Acknowledgments:
Research supported by A.R.C. Grant A69700321
Author information:
School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW 2308, Australia
george@frey.newcastle.edu.au
 