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This paper is motivated by the question of whether nonzero      L 2 {L^2}      -harmonic differentials exist on coverings of a Riemann surface of genus      ⩾  2 \geqslant 2      . Our approach will be via an analogue of the de Rham theorem. Some results concerning the invariance of      L 2 {L^2}      -homology and the intersection number of      L 2 {L^2}      -cycles are demonstrated. A growth estimate for triangulations of planar coverings of the two-holed torus is derived. Finally, the equivalence between the existence of      L 2 {L^2}      -harmonic one-cycles and the transience of random walks on a planar surface is shown.

𝐿²-cohomology of noncompact surfaces