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An algorithmically unsolvable problem in analysis

Authors: A. Lenard and J. Stillwell
Journal: Proc. Amer. Math. Soc. 88 (1983), 129-130
MSC: Primary 58G25; Secondary 03D35
MathSciNet review: 691292
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Abstract: The decision problem of distinguishing between the cases when the Laplace-Beltrami operator on the covering space of a compact manifold has 0 in its spectrum or is bounded away from 0 is algorithmically unsolvable in any class of manifolds that includes all $ 4$-dimensional ones. The proof depends on a result of Brooks connecting the spectrum with the amenability of the fundamental group.

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  • [1] J. Stillwell, The word problem and the isomorphism problem for groups, Bull. Amer. Math. Soc. 6 (1982), 33-56. MR 634433 (82m:20039)
  • [2] R. C. Lyndon and P. E. Schupp, Combinatorial group theory, Ergeb. Math. Grenzgeb., No. 89, Springer-Verlag, Berlin, 1977. MR 0577064 (58:28182)
  • [3] R. Brooks, Amenability and the spectrum of the Laplacian, Bull. Amer. Math. Soc. 6 (1982), 87-79. MR 634438 (83f:58076)
  • [4] F. P. Greenleaf, Invariant means on topological groups, Van Nostrand Math. Studies No. 16, Van Nostrand, Princeton, N. J., 1969. MR 0251549 (40:4776)
  • [5] W. W. Boone, W. Haken and V. Poenaru, On recursively unsolvable problems in topology and their classification, Contributions to Mathematical Logic (H. A. Schmidt, K. Schutte, H. J. Thiele, editors), North-Holland, Amsterdam, 1968. MR 0263090 (41:7695)
  • [6] W. S. Massey, Algebraic topology: An introduction, Harcourt, Brace & World, New York, 1967. MR 0211390 (35:2271)

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Keywords: Unsolvability, Laplacian on manifolds, spectrum on Laplacian, amenability, Markov property
Article copyright: © Copyright 1983 American Mathematical Society

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