Willis Theory via Graphs

TL;DR

This paper introduces a geometric framework for analyzing endomorphisms of totally disconnected locally compact groups, providing new interpretations, procedures, and theorems related to tidy subgroups and the scale function.

Contribution

It develops a geometric approach to tidy subgroups and the scale function, including a new tidying procedure, a proof of tidiness equivalence, and an endomorphism version of the Baumgartner-Willis tree theorem.

Findings

Geometric interpretation of tidy subgroups and the scale function

A new geometric tidying procedure for endomorphisms

An endomorphism version of the Baumgartner-Willis tree representation theorem

Abstract

We study the scale and tidy subgroups of an endomorphism of a totally disconnected locally compact group using a geometric framework. This leads to new interpretations of tidy subgroups and the scale function. Foremost, we obtain a geometric tidying procedure which applies to endomorphisms as well as a geometric proof of the fact that tidiness is equivalent to being minimizing for a given endomorphism. Our framework also yields an endomorphism version of the Baumgartner-Willis tree representation theorem. We conclude with a construction of new endomorphisms of totally disconnected locally compact groups from old via HNN-extensions.