Central theorems for cohomologies of certain solvable groups

Kasuya, Hisashi

doi:10.1090/tran/6837

Central theorems for cohomologies of certain solvable groups
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by Hisashi Kasuya PDF

Trans. Amer. Math. Soc. 369 (2017), 2879-2896 Request permission

Abstract:

We show that the group cohomology of torsion-free virtually polycyclic groups and the continuous cohomology of simply connected solvable Lie groups can be computed by the rational cohomology of algebraic groups. Our results are generalizations of certain results on the cohomology of solvmanifolds and infra-solvmanifolds. Moreover as an application of our results, we give a new proof of the surprising cohomology vanishing theorem given by Dekimpe-Igodt.

References

Donu Arapura and Madhav Nori, Solvable fundamental groups of algebraic varieties and Kähler manifolds, Compositio Math. 116 (1999), no. 2, 173–188. MR 1686777, DOI 10.1023/A:1000879906578
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
Oliver Baues and Fritz Grunewald, Automorphism groups of polycyclic-by-finite groups and arithmetic groups, Publ. Math. Inst. Hautes Études Sci. 104 (2006), 213–268. MR 2264837, DOI 10.1007/s10240-006-0003-3
O. Baues and B. Klopsch, Deformations and rigidity of lattices in solvable Lie groups, J. Topol. 6 (2013), no. 4, 823–856. MR 3145141, DOI 10.1112/jtopol/jtt016
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
Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York-Berlin, 1982. MR 672956
Kenneth S. Brown, Homological criteria for finiteness, Comment. Math. Helv. 50 (1975), 129–135. MR 376820, DOI 10.1007/BF02565740
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
A. Borel and N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, 2nd ed., Mathematical Surveys and Monographs, vol. 67, American Mathematical Society, Providence, RI, 2000. MR 1721403, DOI 10.1090/surv/067
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
Nick Dungey, A. F. M. ter Elst, and Derek W. Robinson, Analysis on Lie groups with polynomial growth, Progress in Mathematics, vol. 214, Birkhäuser Boston, Inc., Boston, MA, 2003. MR 2000440, DOI 10.1007/978-1-4612-2062-6
Richard M. Hain, The Hodge de Rham theory of relative Malcev completion, Ann. Sci. École Norm. Sup. (4) 31 (1998), no. 1, 47–92 (English, with English and French summaries). MR 1604294, DOI 10.1016/S0012-9593(98)80018-9
Richard Hain, Remarks on non-abelian cohomology of proalgebraic groups, J. Algebraic Geom. 22 (2013), no. 3, 581–598. MR 3048547, DOI 10.1090/S1056-3911-2013-00598-6
G. Hochschild, Cohomology of algebraic linear groups, Illinois J. Math. 5 (1961), 492–519. MR 130901
G. Hochschild, Introduction to affine algebraic groups, Holden-Day, Inc., San Francisco, Calif.-Cambridge-Amsterdam, 1971. MR 0277535
G. Hochschild and G. D. Mostow, Cohomology of Lie groups, Illinois J. Math. 6 (1962), 367–401. MR 147577
G. Hochschild and J.-P. Serre, Cohomology of group extensions, Trans. Amer. Math. Soc. 74 (1953), 110–134. MR 52438, DOI 10.1090/S0002-9947-1953-0052438-8
Hisashi Kasuya, Minimal models, formality, and hard Lefschetz properties of solvmanifolds with local systems, J. Differential Geom. 93 (2013), no. 2, 269–297. MR 3024307
Jens Carsten Jantzen, Representations of algebraic groups, Pure and Applied Mathematics, vol. 131, Academic Press, Inc., Boston, MA, 1987. MR 899071
Larry A. Lambe and Stewart B. Priddy, Cohomology of nilmanifolds and torsion-free, nilpotent groups, Trans. Amer. Math. Soc. 273 (1982), no. 1, 39–55. MR 664028, DOI 10.1090/S0002-9947-1982-0664028-2
John McCleary, A user’s guide to spectral sequences, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 58, Cambridge University Press, Cambridge, 2001. MR 1793722
G. D. Mostow, Fully reducible subgroups of algebraic groups, Amer. J. Math. 78 (1956), 200–221. MR 92928, DOI 10.2307/2372490
G. D. Mostow, Cohomology of topological groups and solvmanifolds, Ann. of Math. (2) 73 (1961), 20–48. MR 125179, DOI 10.2307/1970281
Katsumi Nomizu, On the cohomology of compact homogeneous spaces of nilpotent Lie groups, Ann. of Math. (2) 59 (1954), 531–538. MR 64057, DOI 10.2307/1969716
È. B. Vinberg (ed.), Lie groups and Lie algebras, III, Encyclopaedia of Mathematical Sciences, vol. 41, Springer-Verlag, Berlin, 1994. Structure of Lie groups and Lie algebras; A translation of Current problems in mathematics. Fundamental directions. Vol. 41 (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1990 [ MR1056485 (91b:22001)]; Translation by V. Minachin [V. V. Minakhin]; Translation edited by A. L. Onishchik and È. B. Vinberg. MR 1349140, DOI 10.1007/978-3-662-03066-0
E. B. Vinberg (ed.), Lie groups and Lie algebras. II, Encyclopaedia of Mathematical Sciences, vol. 21, Springer-Verlag, Berlin, 2000. Discrete subgroups of Lie groups and cohomologies of Lie groups and Lie algebras; A translation of Current problems in mathematics. Fundamental directions. Vol. 21 (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform. (VINITI), Moscow, 1988 [ MR0968444 (89f:22001)]; Translated by John Danskin; Translation edited by A. L. Onishchik and E. B. Vinberg. MR 1756406
M. S. Raghunathan, Discrete subgroups of Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68, Springer-Verlag, New York-Heidelberg, 1972. MR 0507234
Claire Voisin, Hodge theory and complex algebraic geometry. I, Cambridge Studies in Advanced Mathematics, vol. 76, Cambridge University Press, Cambridge, 2002. Translated from the French original by Leila Schneps. MR 1967689, DOI 10.1017/CBO9780511615344