Splitting quasi-transitive graphs

• Applicant: Dr. Matthias Hamann
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 549406527

In this project, we want to decompose quasi-transitive locally finite graphs into building blocks of a simpler structure. More precisely, we want to obtain a splitting result for one-ended quasi-transitive locally finite graphs with additional assumptions, where the splitting takes place over two-ended quasi-transitive graphs, and we want to prove other close by results. Note that the latter is a well-understood class of quasi-transitive locally finite graphs, one of the simplest ones of infinite such graphs. We aim at tree amalgamations over such graphs or (from another point of view) at canonical tree-decompositions with such graphs as adhesion sets. Of course, if we have done this splitting once, we may ask if the factors satisfy the assumptions again and then split them, too, if possible, and so on. If we end up with only factors that are not splittable, then we obtain the notion of accessibility, otherwise inaccessibility. We will define a notion of graphs of graphs similar to graphs of groups and discuss its behaviour with respect to accessibility and we want to prove a structural result for them similar to the fundamental theorem of Bass-Serre theory, that allows us to recover our original graph from the informations encoded in the graph of graphs. This whole project embeds into a much larger program to understand quasi-transitive locally finite graphs up to quasi-isometries, similar to Gromov’s program to classify finitely generated groups up to quasi-isometry. It is important to note that quasi-transitive locally finite graphs have much more variety in their quasi-isometry classes than finitely generated groups: this follows from an answer to a question of Woess which showed that there are quasi-transitive locally finite graphs that are not quasi-isometric to finitely generated groups. Additionally, we will look at the situation in digraphs and seek for more understanding of the ends of digraphs and for a canonical decomposition that distinguishes the ends of digraphs. We will also look at what such a decomposition result for digraphs implies for semigroups.

• DFG Programme: Research Grants