The DOI matches its Crossref record.
The fundamental group and the spectrum of the laplacian
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Claims the paper makes
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
In the citing paper
“Because the mapping class group is non-amenable, we expect that the spectrum of the Weil-Petersson Laplacian on Teichm¨ uller space is bounded away from zero.”
From Lorentz spacetimes of constant curvature (2007) · cites this paper as [45]
In the citing paper
“A covering of a compact Riemannian manifold with non-amenable deck group has spectrum bounded away from zero, by Brooks's theorem [45]”
From Lorentz spacetimes of constant curvature (2007) · cites this paper as [45]
Citations
Cites
No references could be resolved yet.
Research works printed but not linked: 18
- W. K. Allard,On the First Variation of a Varifold, Ann. Math.95 (1972), 417–491. · DOI
- R. Brooks,Exponential Growth and the Spectrum of the Laplacian, Proc. AMS82 (1981), 473–477. · DOI
- —,Some Remarks on Bounded Cohomology, Ann. Math. Studies97 (1980), 53–63.
- J. Cheeger,A Lower Bound for the Smallest Eigenvalue of the Laplacian, in Gunning,Problems in Analysis (1970), 195–199. · DOI
- J. Cheeger andS. T. Yau,A Lower Bound for the Heat Kernel, preprint.
- S. Y. Cheng andS. T. Yau,Differential Equations on Riemannian Manifolds and their Geometric Applications, Comm. Pure and Appl. Math.,28 (1975), 333–354. · DOI
- H. Donnelly,Asymptotic Expansions for the Compact Quotients of Properly Discontinuous Group Actions, Ill. J. of Math.23 (1979), 485–496. · DOI
- P. Eberlein,Some Properties of the Fundamental Group of a Fuchsian Manifold, Inventiones Math.19 (1973), 5–13. · DOI
- F. Eisenberg, Ph.D. thesis, Stony Brook (1979).
- M. Hirsch andW. Thurston,Foliated Bundles, Invariant Measures, and Flat Manifolds, Ann. Math.101 (1975), 369–390. · DOI
- H. Kesten,Full Banach Mean Values on Countable Groups, Math. Scand.7 (1959), 146–156. · DOI
- H. P. McKean,An Upper Bound to the Spectrum of Δ on a Manifold of Negative Curvature, J. Diff. Geometry4 (1970), 359–366. · DOI
- J. Milnor,A Note on Curvature and Fundamental Group, J. Diff. Geom.2 (1968), 1–7. · DOI
- J. Tits,Free Subgroups in Linear Groups, J. Algebra20 (1972), 250–270. · DOI · Paper page
- H. Federer,Geometric Measure Theory, Springer Verlag (1969).
- E. Følner,On groups with Full Banach Mean Value, Math. Scand.3 (1955), 243–254. · DOI
- H. B. Lawson,Minimal Varieties in Real and Complex Geometry, Séminaire de Mathématiques Superieures, vol. 57, Université de Montréal, 1974.
- S. T. Yau,Isoperimetric Constants and the First Eigenvalue of a Compact Riemannian Manifold, Ann. Sci. Éc. Norm. Sup. (1975) 487–507. · DOI
Cited by
Lorentz spacetimes of constant curvature
Mess, G. (2007) · Geometriae Dedicata · doi:10.1007/s10711-007-9155-7 · Cited by 278
Lorentz spacetimes of constant curvature
Mess, G. (2007) · arXiv · Cited by 5
Metadata
- Authors
- Brooks, R.
- Venue
- Commentarii Mathematici Helvetici
- Published
- 1981-12
- Volume
- 56
- Issue
- 1
- Pages
- 581-598
- ISSN
- 0010-2571
- 1420-8946
- Cited by
- 99
- References
- 0
- Language
- en
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
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How to cite
Cite CoChat for the checks on this page, and the paper for its own findings.
- This page
- CoChat. (2026). Citation check: The fundamental group and the spectrum of the laplacian. https://app.cochat.ai/papers/10.1007/bf02566228
- The paper
- Brooks, R. (1981). The fundamental group and the spectrum of the laplacian. Commentarii Mathematici Helvetici. https://doi.org/10.1007/bf02566228
Data sources
Bibliographic data: Semantic Scholar, OpenAlex, Crossref.
Contains information from the Semantic Scholar Academic Graph Datasets, made available under the ODC Attribution License (ODC-BY 1.0).
Semantic Scholar’s data is described in: Kinney, R., et al. (2023). The Semantic Scholar Open Data Platform. arXiv:2301.10140.
