CoChat
Sign Up
Cited by 266Geometric and Algebraic TopologyAdvanced Differential Geometry ResearchAdvanced Algebra and Geometry

On fundamental groups of complete affinely flat manifolds

Open access
Milnor, John·1977·Advances in Mathematics
DOI 10.1016/0001-8708(77)90004-4
Open this paper in CoChat — free accountOpen PDF

The claims and statistics of this paper haven’t been checked yet.

Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.

A few free checks a day · papers already verified open instantly · results are public

Integrity signals

Registry

Crossref · 10.1016/0001-8708(77)90004-4

Cites

1 of 1 references verified0 citation errors · 0 not checkable

Statistics

Claims the paper makes

The claims and statistics of this paper haven’t been checked yet.

Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.

A few free checks a day · papers already verified open instantly · results are public

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?

  • Given f : π1S → SO(2,1)0 and M = L+/f(π1S) as in Proposition 2, and any bounded neighbourhood U of 0 in H1(π1S,R2+1), there exists C such that, letting M_C = { (x,y,t) : -t^2 + x^2 + y^2 ≤ -C }/f(π1S), there exists a family of flat Lorentz metrics on M_C parametrized by U such that the spacetime M_C(u) corresponding to u ∈ U has holonomy ρ : π1M_C → ISO(2,1) such that qρ = f and ρ(γ)x = f(γ)x + tγ

    “Proposition 3. Given f : π 1 S → SO (2 , 1) 0 and M = L + /f ( π 1 S ) as in proposition 2, and any bounded neighbourhood U of 0 in H 1 ( π 1 S, R 2+1 ) there exists C such that, letting M C = { ( x, y, t ) : -t 2 + x 2 + y 2 ≤ -C } /f ( π 1 S ) , there exists a family of flat Lorentz metrics on M C parametrized by U”

    From Lorentz spacetimes of constant curvature (2007) · cites this paper as [15]

    Split into 2 atoms — 0 confirmed, 1 partially supported, 0 disputed, 0 misattributed, 1 unverifiable. Expand the claim to see per-atom verdicts.

    Experimental — accuracy not yet measured

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

The self-check needs the paper’s full text.

Cites

Flagged first, then by citations

Research works printed but not linked: 20

  1. L. AUSLANDER, Examples of locally afhne spaces, Ann. of Math. 64 (1956), 255-259.
  2. L. AUSLANDER, On a problem of Phillip Hall, Ann. of Math. 86 (1967), 112-l 16.
  3. L. AUSLANDER AND F. E. A. JOHNSON, On a conjecture of C. T. C. Wall, 1. London Math. Sot. 14 (1976), 331-332.
  4. L. BIEBERBACH, tfber die Bewegungsgruppen der Euklidischen Raume, Math. Ann. 70 (1911), 297-336. (Also 72 (1912), 400-412.)
  5. J. CHEEGER AND D. GROMOLL, On the structure of complete manifolds of nonnegative curvature, Ann. of Math. 96 (1972), 413-443.
  6. IO. J. DIXMIER AND W. G. LISTER, Derivations of nilpotent Lie algebras, PYOC. Amer. Math. Sot. 8 (1957), 155-158.
  7. H. FURSTENBERG, A Poisson formula for semi-simple Lie groups, Ann. of Math. 77 (1963), 335-386.
  8. S. HELGASON, "Differential Geometry and Symmetric Spaces," Academic Press, New York, 1962.
  9. N. JACOBSON, A note on automorphisms and derivations of Lie algebras, PYOC. Amer. Math. Sot. 6 (1955), 281-283.
  10. B. KOSTANT AND D. SULLIVAN, The Euler characteristic of an affine space form is zero, Bull. Amer. Math. Sot. 81 (1975), 937-938.
  11. J. MILNOR, Hilbert's problem 18: on crystalographic groups, fundamental domains, and on sphere packing, in "Proceedings Symposia Pure Math.," Vol. 28, pp. 491-506, Amer. Math. Sot., Providence, R.I., 1976.
  12. G. D. MOSTOW, On the fundamental group of homogeneous spaces, Ann. of Math. 66 (1957), 249-255.
  13. N. W. RICKERT, Amenable groups and groups with the fixed point property, Trans. Amer. Math. Sot. 127 (1967), 221-232.
  14. J. SCHEUNEMAN, Examples of compact locally afline spaces, Bull. Amer. Math. Sot. 77 (1971), 589-592.
  15. J. SCHEUNEMAN, Affine structures on three-step nilpotent Lie algebras, Proc. Amer. Math. Sot. 46 (1974), 451-454.
  16. J. SMILLIE, "Flat Manifolds with Non-Zero Euler Characteristic," preprint, Uni- versity of Chicago, 1976.
  17. R. SWAN, Representations of polycyclic groups, Proc. Amer. Math. Sot. 18 (1967), 573-574.
  18. J. TITS, Free subgroups in linear groups, J. Algebra 20 (1972), 250-270.
  19. J. WOLF, "Spaces of Constant Curvature," McGraw-Hill, New York, 1967, and Publish or Perish, Boston, Mass., 1974.
  20. M. HIRSCH AND W. THURSTON, Foliated bundles, invariant measures and flat mani- folds, Ann. of Math. 101 (1975), 369-390.

Cited by

Sorted by citations

Authors
Milnor, J.
Affiliations
Institute for Advanced Study (US)
Corresponding author
Milnor, John
Venue
Advances in Mathematics
Publisher
Elsevier BV
Published
1977-08
Volume
25
Issue
2
Pages
178-187
ISSN
0001-8708 (ISSN-L)
DOI
10.1016/0001-8708(77)90004-4
OpenAlex
W2048250798
Cited by
266
References
1
Citations by year
2026: 6 · 2025: 7 · 2024: 10 · 2023: 4 · 2022: 5 · 2021: 16 · 2020: 5 · 2019: 10 · 2018: 5 · 2017: 7 · 2016: 8 · 2015: 9 · 2014: 6 · 2013: 3 · 2012: 5
FWCI
5.88
Type
journal article
Language
en
Open access
Licence
Locations
Advances in Mathematics (publishedVersion)
Funders
University of Warwick
Topics
  • Physical Sciences › Mathematics › Geometry and Topology › Geometric and Algebraic Topology
  • Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Advanced Differential Geometry Research
  • Physical Sciences › Mathematics › Mathematical Physics › Advanced Algebra and Geometry
Keywords
fundamental group, affine manifolds, torsion-free groups

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.

Statistics and arithmetic are recomputed, not judged.

Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.

What counts as a claim ›How we measure confidence ›How we check this ›

Are you an author, or do you see a problem with this page? Request a review