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

Let \alpha be an automorphism of the totally disconnected group G. The compact open subgroup, V, if G is 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 H 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 H the analogues of eigenvalues.