Bobby Miraftab (Carleton University)

Title: From Geometric Group Theory to Structural Graph Theory

Abstract: Geometric group theory extracts information about a finitely generated group from the large-scale geometry of its Cayley graph. Coarse invariants such as the number of ends, hyperbolicity, and growth have powerful consequences, most famously Stallings’ theorem, which detects splittings of groups from their ends, and the cubulating groups via actions on CAT(0) cube complexes. A natural generalization of Cayley graphs is given by transitive graphs, i.e. graphs whose automorphism group acts transitively on vertices. Many coarse-geometric notions (ends, hyperbolicity, growth) extend naturally to this broader setting, but transferring group-theoretic structure theorems to transitive graphs requires new tools.

In this talk I introduce methods from structural graph theory, especially tree decompositions and median decompositions and explain how they provide a framework for extending key results from geometric group theory to transitive graphs. In particular, I discuss analogues of Stallings-type splitting phenomena and connections to cubulations via median structures.

We will gather in the lounge for teatime after the talk, and then we will go for dinner with Bobby.