Crystallographic actions on contractible algebraic manifolds

Dekimpe, Karel; Petrosyan, Nansen

doi:10.1090/S0002-9947-2014-06160-6

Crystallographic actions on contractible algebraic manifolds
HTML articles powered by AMS MathViewer

by Karel Dekimpe and Nansen Petrosyan PDF

Trans. Amer. Math. Soc. 367 (2015), 2765-2786 Request permission

Abstract:

We study properly discontinuous and cocompact actions of a discrete subgroup $\Gamma$ of an algebraic group $G$ on a contractible algebraic manifold $X$. We suppose that this action comes from an algebraic action of $G$ on $X$ such that a maximal reductive subgroup of $G$ fixes a point. When the real rank of any simple subgroup of $G$ is at most one or the dimension of $X$ is at most three, we show that $\Gamma$ is virtually polycyclic. When $\Gamma$ is virtually polycyclic, we show that the action reduces to an NIL-affine crystallographic action. Specializing to NIL-affine actions, we prove that the generalized Auslander conjecture holds up to dimension six and give a new proof of the fact that every virtually polycyclic group admits an NIL-affine crystallographic action.

References

Herbert Abels, Grigori A. Margulis, and Grigori A. Soifer, Properly discontinuous groups of affine transformations with orthogonal linear part, C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), no. 3, 253–258 (English, with English and French summaries). MR 1438395, DOI 10.1016/S0764-4442(99)80356-5
Herbert Abels, Properly discontinuous groups of affine transformations: a survey, Geom. Dedicata 87 (2001), no. 1-3, 309–333. MR 1866854, DOI 10.1023/A:1012019004745
Louis Auslander, Bieberbach’s theorems on space groups and discrete uniform subgroups of Lie groups, Ann. of Math. (2) 71 (1960), 579–590. MR 121423, DOI 10.2307/1969945
Louis Auslander, The structure of complete locally affine manifolds, Topology 3 (1964), no. suppl, suppl. 1, 131–139. MR 161255, DOI 10.1016/0040-9383(64)90012-6
Ludwig Bieberbach, Über die Bewegungsgruppen der Euklidischen Räume, Math. Ann. 70 (1911), no. 3, 297–336 (German). MR 1511623, DOI 10.1007/BF01564500
Ludwig Bieberbach, Über die Bewegungsgruppen der Euklidischen Räume (Zweite Abhandlung.) Die Gruppen mit einem endlichen Fundamentalbereich, Math. Ann. 72 (1912), no. 3, 400–412 (German). MR 1511704, DOI 10.1007/BF01456724
Oliver Baues, Infra-solvmanifolds and rigidity of subgroups in solvable linear algebraic groups, Topology 43 (2004), no. 4, 903–924. MR 2061212, DOI 10.1016/S0040-9383(03)00083-1
Yves Benoist, Une nilvariété non affine, J. Differential Geom. 41 (1995), no. 1, 21–52 (French, with English summary). MR 1316552
Yves Benoist and Karel Dekimpe, The uniqueness of polynomial crystallographic actions, Math. Ann. 322 (2002), no. 3, 563–571. MR 1895707, DOI 10.1007/s002080200005
J. Bochnak, M. Coste, and M. Roy, Real algebraic geometry, Springer-Verlag, New York 1998.
Armand Borel, Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. MR 1102012, DOI 10.1007/978-1-4612-0941-6
Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York-Berlin, 1982. MR 672956, DOI 10.1007/978-1-4684-9327-6
Dietrich Burde, Karel Dekimpe, and Sandra Deschamps, The Auslander conjecture for NIL-affine crystallographic groups, Math. Ann. 332 (2005), no. 1, 161–176. MR 2139256, DOI 10.1007/s00208-004-0623-1
Virginie Charette, Todd Drumm, William Goldman, and Maria Morrill, Complete flat affine and Lorentzian manifolds, Geom. Dedicata 97 (2003), 187–198. Special volume dedicated to the memory of Hanna Miriam Sandler (1960–1999). MR 2003697, DOI 10.1023/A:1023680928912
Lawrence J. Corwin and Frederick P. Greenleaf, Representations of nilpotent Lie groups and their applications. Part I, Cambridge Studies in Advanced Mathematics, vol. 18, Cambridge University Press, Cambridge, 1990. Basic theory and examples. MR 1070979
Karel Dekimpe, Polynomial crystallographic actions on the plane, Geom. Dedicata 93 (2002), 47–56. MR 1934685, DOI 10.1023/A:1020337728898
Karel Dekimpe, Any virtually polycyclic group admits a NIL-affine crystallographic action, Topology 42 (2003), no. 4, 821–832. MR 1958530, DOI 10.1016/S0040-9383(02)00030-7
Karel Dekimpe and Paul Igodt, Polycyclic-by-finite groups admit a bounded-degree polynomial structure, Invent. Math. 129 (1997), no. 1, 121–140. MR 1464868, DOI 10.1007/s002220050160
Todd A. Drumm and William M. Goldman, Complete flat Lorentz $3$-manifolds with free fundamental group, Internat. J. Math. 1 (1990), no. 2, 149–161. MR 1060633, DOI 10.1142/S0129167X90000101
Todd A. Drumm and William M. Goldman, The geometry of crooked planes, Topology 38 (1999), no. 2, 323–351. MR 1660333, DOI 10.1016/S0040-9383(98)00016-0
David Fried and William M. Goldman, Three-dimensional affine crystallographic groups, Adv. in Math. 47 (1983), no. 1, 1–49. MR 689763, DOI 10.1016/0001-8708(83)90053-1
M. Gromov, Almost flat manifolds, J. Differential Geometry 13 (1978), no. 2, 231–241. MR 540942, DOI 10.4310/jdg/1214434488
Kyung Bai Lee and Frank Raymond, Rigidity of almost crystallographic groups, Combinatorial methods in topology and algebraic geometry (Rochester, N.Y., 1982) Contemp. Math., vol. 44, Amer. Math. Soc., Providence, RI, 1985, pp. 73–78. MR 813102, DOI 10.1090/conm/044/813102
L. Magnin, Adjoint and trivial cohomology tables for indecomposable nilpotent Lie algebras of dimension $\leq 7$ over $\mathbb {C}$, Electronic Book, 1995.
G. A. Margulis, Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 17, Springer-Verlag, Berlin, 1991. MR 1090825, DOI 10.1007/978-3-642-51445-6
Geoffrey Mess, Lorentz spacetimes of constant curvature, Geom. Dedicata 126 (2007), 3–45. MR 2328921, DOI 10.1007/s10711-007-9155-7
John Milnor, On fundamental groups of complete affinely flat manifolds, Advances in Math. 25 (1977), no. 2, 178–187. MR 454886, DOI 10.1016/0001-8708(77)90004-4
A. Onishchik, and E. Vinberg, Lie groups and Lie algebras, III, Encyclopedia of Mathematical Sciences, 41, Springer-Verlag, 1994.
M. S. Raghunathan, Discrete subgroups of Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68, Springer-Verlag, New York-Heidelberg, 1972. MR 0507234, DOI 10.1007/978-3-642-86426-1
Maxwell Rosenlicht, On quotient varieties and the affine embedding of certain homogeneous spaces, Trans. Amer. Math. Soc. 101 (1961), 211–223. MR 130878, DOI 10.1090/S0002-9947-1961-0130878-0
Ernst A. Ruh, Almost flat manifolds, J. Differential Geometry 17 (1982), no. 1, 1–14. MR 658470
I. R. Shafarevich, Basic algebraic geometry, Die Grundlehren der mathematischen Wissenschaften, Band 213, Springer-Verlag, New York-Heidelberg, 1974. Translated from the Russian by K. A. Hirsch. MR 0366917, DOI 10.1007/978-3-642-96200-4
G. A. Soĭfer, Affine crystallographic groups, Algebra and analysis (Irkutsk, 1989) Amer. Math. Soc. Transl. Ser. 2, vol. 163, Amer. Math. Soc., Providence, RI, 1995, pp. 165–170. MR 1331393, DOI 10.1090/trans2/163/14
G. Tomanov, The virtual solvability of the fundamental group of a generalized Lorentz space form, J. Differential Geom. 32 (1990), no. 2, 539–547. MR 1072918, DOI 10.4310/jdg/1214445319
Edward N. Wilson, Isometry groups on homogeneous nilmanifolds, Geom. Dedicata 12 (1982), no. 3, 337–346. MR 661539, DOI 10.1007/BF00147318
Hans Zassenhaus, Über einen Algorithmus zur Bestimmung der Raumgruppen, Comment. Math. Helv. 21 (1948), 117–141 (German). MR 24424, DOI 10.1007/BF02568029