Discrete (P)-closed Groups Acting On Trees

TL;DR

This paper characterizes when groups acting on trees with property (P), described by local action diagrams, are discrete, linking group properties to combinatorial structures.

Contribution

It provides necessary and sufficient conditions on local action diagrams for the associated groups to be discrete, advancing understanding of their structure.

Findings

Conditions for discreteness of groups from local action diagrams

Connection between group properties and combinatorial structures

Extension of Reid-Smith parametrization to discreteness

Abstract

Reid-Smith recently parametrised groups acting on trees with Tits' independence property (P) using graph-based combinatorial structures known as local action diagrams. Properties of the acting (topological) group, such as being locally compact, compactly generated or simple, are reflected in its local action diagram. In this article we provide necessary and sufficient conditions on the local action diagram for the associated group to be discrete.