A linear isoperimetric inequality for the punctured Euclidean plane

It follows from a general theorem of Bonk and Eremenko that closed plane curves which are contractible in the complement to the integral lattice satisfy a linear isoperimetric inequality. We give an alternative proof of this fact. Our approach is based on a non-standard combinatorial isoperimetric inequality which requires a refinement of the small cancellation theory. We present an application of the isoperimetric inequality for the punctured plane to Hamiltonian dynamics. Combining it with methods of symplectic topology we show that every non-identical Hamiltonian diffeomorphism of the 2-torus has at least linear asymptotic growth of the differential.