The Scale of ($P$)-closed Groups Acting On Trees

Abstract

Reid--Smith parametrised ($P$)-closed groups acting on trees using graph-based combinatorial structures known as local action diagrams. Properties of the acting (topological) group, such as being locally compact, compactly generated, discrete or simple, are reflected in its local action diagram. In this article, we describe the translations of ($P$)-closed groups and their axes in terms of local action diagrams. As applications, we determine the scale function of ($P$)-closed groups and characterise unimodular as well as uniscalar ($P$)-closed groups. The latter provides one possible answer to a question of Thomas Weigel.