The scale function and tidy subgroups

TL;DR

This paper introduces the scale function and tidy subgroups in the structure theory of totally disconnected locally compact groups, illustrating their properties and implications such as the closure of periodic elements and restrictions on ergodic automorphisms.

Contribution

It provides an accessible exposition of key tools in the structure theory of totally disconnected groups, including new proofs of fundamental properties.

Findings

Periodic elements form a closed set in such groups.

Non-compact groups cannot have ergodic automorphisms.

The paper clarifies properties and examples of the scale function and tidy subgroups.

Abstract

This exposition article arose from two talks given during the Oberwolfach Arbeitsgemeinschaft on Totally Disconnected Groups in October 2014. This is an introduction to the structure theory of totally disconnected locally compact groups initiated by Willis in 1994. The two main tools in this theory are the scale function and tidy subgroups, for which we present several properties and examples. As an illustration of this theory, we give a proof of the fact that the set of periodic elements in a totally disconnected locally compact group is always closed, and that such a group cannot have ergodic automorphisms as soon as it is non-compact.