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Graphs of group actions and group actions on trees

Florian Lehner, Christian Lindorfer, Rögnvaldur G. Möller, Wolfgang Woess

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.

Graphs of group actions and group actions on trees

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.

Paper Structure

This paper contains 35 sections, 45 theorems, 18 equations, 6 figures.

Table of Contents

  1. Preliminaries
  2. Graphs
  3. Group actions
  4. Topological groups and the permutation topology
  5. Wreath products
  6. Graphs of group actions and their universal groups
  7. Graphs of group actions
  8. Scaffoldings
  9. The universal group of a graph of group actions
  10. Existence of scaffoldings and universal groups
  11. The augmented base digraph
  12. Star-covering trees
  13. Canonical scaffoldings
  14. Acceptable isomorphisms and the universal group
  15. The extension property
  16. ...and 20 more sections

Key Result

Lemma 1.2

Let $G$ be a group acting on a set $X$. Assume $G$ is closed in the permutation topology. Assume also that all suborbits (the orbits of stabilizers of points) are finite. Then:

Figures (6)

  • Figure 1: Diagram describing isomorphisms of group actions.
  • Figure 2: If $\gamma \in G(t(a))$, then $\Psi$ in this commutative diagram is an $a$-adhesion map and $\Psi(Y(a))$ is an $a$-adhesion set.
  • Figure 3: The compatibilty conditions for a legal colouring at the arc $\llbracket a\rrbracket \in \mathrm A T_\Sigma$ with $t(\llbracket a\rrbracket) = \llbracket v\rrbracket$, $o(\llbracket a\rrbracket) = \llbracket w\rrbracket$ say that the above diagram commutes. The arrow marked with $\mathop{\overline{\,\cdot\,}}$ denotes the action of reversing an arc and the arrow labelled by $=$ is the identity map.
  • Figure 4: Commutative diagram for the local action $g_{\llbracket v \rrbracket}$ of $g \in \mathrm{Aut}_{\mathrm{sc}}(\Sigma)$ at $\llbracket v \rrbracket \in \mathrm V T_{\Sigma}$.
  • Figure 5: Commutative diagram defining the local map $f_{[v]}$ of $f$ at $[v]\in \mathrm V T_{\Sigma}$.
  • ...and 1 more figures

Theorems & Definitions (113)

  • Definition 1.1
  • Lemma 1.2
  • Remark 1.3
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Definition 2.7
  • ...and 103 more