Graphs of group actions and group actions on trees

TL;DR

This paper develops a new framework for constructing and analyzing group actions on trees, generalizing existing theories and enabling systematic local-to-global group action constructions.

Contribution

It introduces graphs of group actions as an analogue to graphs of groups, unifying and extending Bass-Serre theory and local action diagrams.

Findings

Established uniqueness and universality of the constructed groups.

Unified and generalized existing constructions like graphs of groups.

Enabled efficient construction of groups with specific local properties.

Abstract

Bass-Serre theory provides a powerful framework for studying group actions on trees. While extremely effective for structural questions in group theory, it is less suited to the systematic construction of group actions with prescribed local behaviour. Motivated by local-to-global constructions such as the Burger-Mozes universal groups and local action diagrams, we develop an analogue of Bass-Serre theory for group actions. The central object of study in our are graphs of group actions, combinatorial structures similar to graphs of groups from Bass-Serre theory, encoding compatible local permutation actions on a base graph. From these we can construct groups which act on tree-like graphs called scaffoldings and hence also on trees. We prove uniqueness and universality results for the resulting groups and show that our framework unifies and generalises (among other known constructions) both graphs of groups and local action diagrams. Remarkably, we are able to encapsulate the full generality of the former while still allowing for efficient construction of groups with certain local properties like in the latter.