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New York Journal of Mathematics 10 (2004), 1-35.

Tidy subgroups for commuting automorphisms of totally disconnected groups: An analogue of simultaneous triangularisation of matrices

George A. Willis

Published: January 7, 2004
Keywords: locally compact group, scale function, tidy subgroup, modular function, automorphism
Subject: Primary 22D05; Secondary 22D45, 20E25, 20E36


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.

Research supported by A.R.C. Grant A69700321

Author information:
School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW 2308, Australia