Short proof shows infinite diameter for surface diffeomorphisms.
problem Infinite diameter of surface diffeomorphisms group.
method Short proof using -diameter concept.
result Infinite -diameter of group.
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Infinite diameter proved for contractible loops space.
problem Infinite diameter of contractible loops space in a compact surface.
method New functionals on normed groups, more general than quasi-morphisms.
result Proved infinite diameter of contractible loops space.
We show that for each the -metric on the group of area-preserving diffeomorphisms of the two-sphere has infinite diameter. This solves the last open case of a conjecture of Shnirelman from 1985. Our methods extend to yield stronger results on the large-scale geometry of the corresponding metric space, …