Infinitesimal rigidity for the action of 𝑆𝐿(𝑛,𝑍) on 𝑇ⁿ

Lewis, James W.

doi:10.1090/S0002-9947-1991-1058434-X

Infinitesimal rigidity for the action of $\textrm {SL}(n,\textbf {Z})$ on $\textbf {T}^ n$
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by James W. Lewis PDF

Trans. Amer. Math. Soc. 324 (1991), 421-445 Request permission

Abstract:

Let $\Gamma = {\mathbf {SL}}(n,\mathbb {Z})$ or any subgroup of finite index. Then the action of $\Gamma$ on ${\mathbb {T}^n}$ by automorphisms is infinitesimally rigid for $n \ge 7$, i.e., the cohomology ${\text {H}^1}(\Gamma ,\operatorname {Vec} ({\mathbb {T}^n})) = 0$, where $\operatorname {Vec} ({\mathbb {T}^n})$ denotes the module of ${C^\infty }$ vector fields on ${\mathbb {T}^n}$.

References

Armand Borel, Stable real cohomology of arithmetic groups, Ann. Sci. École Norm. Sup. (4) 7 (1974), 235–272 (1975). MR 387496
Armand Borel, Stable real cohomology of arithmetic groups. II, Manifolds and Lie groups (Notre Dame, Ind., 1980) Progr. Math., vol. 14, Birkhäuser, Boston, Mass., 1981, pp. 21–55. MR 642850
A. Borel and J.-P. Serre, Corners and arithmetic groups, Comment. Math. Helv. 48 (1973), 436–491. MR 387495, DOI 10.1007/BF02566134
H. Garland and W.-C. Hsiang, A square integrability criterion for the cohomology of arithmetic groups, Proc. Nat. Acad. Sci. U.S.A. 59 (1968), 354–360. MR 228504, DOI 10.1073/pnas.59.2.354
Saunders Mac Lane, Homology, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1975 edition. MR 1344215

Discrete groups of motions of manifolds of non-positive curvature

109

G. A. Margulis, Non-uniform lattices in semisimple algebraic groups, Lie groups and their representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971) Halsted, New York, 1975, pp. 371–553. MR 0422499
Yozô Matsushima and Shingo Murakami, On vector bundle valued harmonic forms and automorphic forms on symmetric riemannian manifolds, Ann. of Math. (2) 78 (1963), 365–416. MR 153028, DOI 10.2307/1970348
G. D. Mostow, Self-adjoint groups, Ann. of Math. (2) 62 (1955), 44–55. MR 69830, DOI 10.2307/2007099
G. D. Mostow, Strong rigidity of locally symmetric spaces, Annals of Mathematics Studies, No. 78, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1973. MR 0385004

Cohomology groups of vector-valued forms on symmetric spaces

Gopal Prasad, Strong rigidity of $\textbf {Q}$-rank $1$ lattices, Invent. Math. 21 (1973), 255–286. MR 385005, DOI 10.1007/BF01418789
M. S. Raghunathan, Discrete subgroups of Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68, Springer-Verlag, New York-Heidelberg, 1972. MR 0507234
M. S. Raghunathan, On the first cohomology of discrete subgroups of semisimple Lie groups, Amer. J. Math. 87 (1965), 103–139. MR 173730, DOI 10.2307/2373227
Atle Selberg, On discontinuous groups in higher-dimensional symmetric spaces, Contributions to function theory (Internat. Colloq. Function Theory, Bombay, 1960) Tata Institute of Fundamental Research, Bombay, 1960, pp. 147–164. MR 0130324
William A. Veech, Periodic points and invariant pseudomeasures for toral endomorphisms, Ergodic Theory Dynam. Systems 6 (1986), no. 3, 449–473. MR 863205, DOI 10.1017/S0143385700003606
E. Vesentini, Lectures on Levi convexity of complex manifolds and cohomology vanishing theorems, Tata Institute of Fundamental Research Lectures on Mathematics, No. 39, Tata Institute of Fundamental Research, Bombay, 1967. Notes by M. S. Raghunathan. MR 0232016
André Weil, On discrete subgroups of Lie groups, Ann. of Math. (2) 72 (1960), 369–384. MR 137792, DOI 10.2307/1970140
André Weil, On discrete subgroups of Lie groups. II, Ann. of Math. (2) 75 (1962), 578–602. MR 137793, DOI 10.2307/1970212
André Weil, Remarks on the cohomology of groups, Ann. of Math. (2) 80 (1964), 149–157. MR 169956, DOI 10.2307/1970495
Robert J. Zimmer, Actions of semisimple groups and discrete subgroups, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 1247–1258. MR 934329
Robert J. Zimmer, Infinitesimal rigidity for smooth actions of discrete subgroups of Lie groups, J. Differential Geom. 31 (1990), no. 2, 301–322. MR 1037402
Robert J. Zimmer, Lattices in semisimple groups and distal geometric structures, Invent. Math. 80 (1985), no. 1, 123–137. MR 784532, DOI 10.1007/BF01388551
Robert J. Zimmer, Lattices in semisimple groups and invariant geometric structures on compact manifolds, Discrete groups in geometry and analysis (New Haven, Conn., 1984) Progr. Math., vol. 67, Birkhäuser Boston, Boston, MA, 1987, pp. 152–210. MR 900826, DOI 10.1007/978-1-4899-6664-3_{6}
Steven Zucker, $L_{2}$ cohomology of warped products and arithmetic groups, Invent. Math. 70 (1982/83), no. 2, 169–218. MR 684171, DOI 10.1007/BF01390727