Hyperbolic groups may be defined as finitely generated groups admitting a linear Dehn function. I wonder whether it is possible to prove most of the classifical properties of hyperbolic groups in this context, probably using van Kampen diagrams. For instance, using this point of view:
Is it possible to prove that a hyperbolic group cannot contain $\mathbb{Z}^2$?
Is it possible to prove that the size of the finite subgroups of a hyperbolic group is bounded?