  Free planar electrons in a uniform magnetic field are shown to possess the
symmetry of area-preserving diffeomorphisms ($W$-infinity algebra).
Intuitively, this is a consequence of gauge invariance, which forces dynamics
to depend only on the flux. The infinity of generators of this symmetry act
within each Landau level, which is infinite-dimensional in the thermodynamical
limit. The incompressible ground states corresponding to completely filled
Landau levels (integer quantum Hall effect) are shown to be infinitely
symmetric, since they are annihilated by an infinite subset of generators. This
geometrical characterization of incompressibility also holds for fractional
fillings of the lowest level (simplest fractional Hall effect) in the presence
of Haldane's effective two-body interactions. Although these modify the
symmetry algebra, the corresponding incompressible ground states proposed by
Laughlin are again symmetric with respect to the modified infinite algebra.
