Small cancellation theory is a tool for constructing finitely generated groups with extreme properties that often serve as counterexamples to conjectures. Graphical small cancellation theory is a generalization of classical small cancellation theory that allows constructions of finitely generated groups with prescribed embedded subgraphs in their Cayley graphs. It has provided such celebrated groups as Gromov's monsters.

In the first part of the talk, I will discuss basic notions and fundamental results of small cancellation theory and present a combinatorial approach to graphical small cancellation theory.

In the second part of the talk, I will show that infinitely presented graphical \(C(7)\)-groups are acylindrically hyperbolic. This has strong implications for these groups: for example, they have free normal subgroups, their reduced \(C^*\)-algebras are simple, and all their asymptotic cones have cut-points. I will moreover present infinitely presented small cancellation groups that have divergence functions with previously unknown behavior.

This is joint work with Alessandro Sisto.