Willis theory via graphs (Q2286354)

Willis' theory made a systematic study of totally disconnected locally compact groups feasible, giving then start to the research interest we now benefit from. In 2015 Willis extended the theory from automorphisms to endomorphisms and, since then, many other mathematicians have worked to generalize well-known results concerning Willis' theory to the more general case of endomorphisms. This paper is part of such an ongoing program. The core is the generalisation of the main results from [\textit{R. G. Möller}, Can. J. Math. 54, No. 4, 795--827 (2002; Zbl 1007.22010)]: the authors obtain a characterization of the main ingredients of Willis' theory for endomorphisms via graphs and permutations. Firstly, they characterize the notions of being tidy above, tidy below and tidy in terms of directed graphs. Subsequently, they develop a geometric tidy procedure: an algorithm which produces a tidy subgroup for a given endomorphism accepting an arbitrary compact open subgroup as input. Once the main ingredients are available in the geometric setup, they are ready to link the existence of tidy subgroups to the value of the scale function an recover the main results of [\textit{G. A. Willis}, Math. Ann. 361, No. 1--2, 403--442 (2015; Zbl 1308.22002)]: being minimizing and being tidy for a given endomorphism are equivalent conditions on compact open subgroups, and the scale of the endomorphism can be detected as out-valency of the root of a directed tree associated to the tidy subgroup. In the remaining part of the paper, the authors use the developed ideas to obtain a tree representation theorem generalizing results from [\textit{U. Baumgartner} and \textit{G. A. Willis}, Isr. J. Math. 142, 221--248 (2004; Zbl 1056.22001)] for the case of automorphisms.