Willis Theory via Graphs

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.

Revised statement and proof of Lemma 5.2. Amended incomplete arguments in the proofs of Lemma 5.4, Proposition 6.3, Lemma 6.4 and Theorem 6.6. General editing and fixing typos