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

TL;DR

This paper explores the structure of automorphisms of totally disconnected groups, establishing an analogy to matrix triangularization by showing commuting automorphisms share a common tidy subgroup, akin to a basis for Jordan form.

Contribution

It demonstrates that commuting automorphisms have a common tidy subgroup and characterizes abelian automorphism groups via invariant tidy subgroups, extending matrix analogy to group automorphisms.

Findings

Commuting automorphisms share a common tidy subgroup.

A group of automorphisms with a common tidy subgroup is abelian modulo invariants.

Subgroups analogous to eigenspaces are identified in the group context.

Abstract

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.