An algorithmically unsolvable problem in analysis

Lenard, A.; Stillwell, J.

doi:10.1090/S0002-9939-1983-0691292-2

An algorithmically unsolvable problem in analysis
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by A. Lenard and J. Stillwell

Proc. Amer. Math. Soc. 88 (1983), 129-130

DOI: https://doi.org/10.1090/S0002-9939-1983-0691292-2

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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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