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Transactions of the American Mathematical Society

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$ L\sp{2}$-cohomology of noncompact surfaces


Author: David R. DeBaun
Journal: Trans. Amer. Math. Soc. 284 (1984), 543-565
MSC: Primary 58A14; Secondary 30F30, 58G32, 60J15
MathSciNet review: 743732
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Abstract: This paper is motivated by the question of whether nonzero $ {L^2}$-harmonic differentials exist on coverings of a Riemann surface of genus $ \geqslant 2$. Our approach will be via an analogue of the de Rham theorem. Some results concerning the invariance of $ {L^2}$-homology and the intersection number of $ {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}$-harmonic one-cycles and the transience of random walks on a planar surface is shown.


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DOI: https://doi.org/10.1090/S0002-9947-1984-0743732-3
Article copyright: © Copyright 1984 American Mathematical Society