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


George A. Willis

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

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

Let α be an automorphism of the totally disconnected group G. The compact open subgroup, V, of G is tidy for α if [α(V') : α(V')∩ V'] is minimised at V, where V' ranges over all compact open subgroups of G. Identifying a subgroup tidy for α 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