Florian Lehner

Groups acting on trees and tree like graphs

Abstract:

Part 1: Invariant tree decompositions and Stallings' theorem

Studying complicated structures by breaking them up into smaller (and usually simpler) parts is a standard approach in mathematics. Decompositions which look tree-like can be particularly powerful as evidenced by beautiful results not just in graph theory, but also in many related fields including formal language theory, computational complexity, probability theory, and group theory.

The focus of this talk is on tree decompositions of graphs which are invariant under the symmetries of the given graph. Classical examples include the block-cut-decomposition and Tutte’s 3-block tree decomposition; higher connectivity analogues were first defined by Dunwoody and Krön in 2009. We will particularly focus on tree decompositions of infinite graphs which are a key ingredient to Krön's combinatorial proof of Stallings’ splitting theorem (a fundamental result in geometric group theory).

Part 2: Group actions defined by local actions

Bass-Serre theory is the perhaps most important tool for studying groups acting on trees. However, its usefulness in constructing group actions with given local properties is limited. To overcome this issue, Reid and Smith recently introduced the theory of local action diagrams which provides a 1-to-1 correspondence between such diagrams and (P)-closed groups acting on trees.

The aim of this talk is to introduce local action diagrams and tree amalgamation (a concept introduced by Mohar which can be seen as an inverse to tree decomposition). Time permitting, we will also discuss graphs of group actions, a notion generalising local action diagrams while retaining many interesting properties of the theory, in particular constructability of groups from given local information.