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Keywords:
nilpotent Lie group; isometric nilmanifolds; normalizer; Lie algebroid; normal subgroupoid system; inner automorphism
nilpotent Lie group; isometric nilmanifolds; normalizer; Lie algebroid; normal subgroupoid system; inner automorphism
Summary:
We study a problem of isometric compact 2-step nilmanifolds ${M}/\Gamma $ using some information on their geodesic flows, where $M$ is a simply connected 2-step nilpotent Lie group with a left invariant metric and $\Gamma $ is a cocompact discrete subgroup of isometries of $M$. Among various works concerning this problem, we consider the algebraic aspect of it. In fact, isometry groups of simply connected Riemannian manifolds can be characterized in a purely algebraic way, namely by normalizers. So, suitable factorization of normalizers and expression of a vector bundle as an associated fiber bundle to a principal bundle, lead us to a general framework, namely groupoids. In this way, drawing upon advanced ingredients of Lie groupoids, normal subgroupoid systems and other notions, not only an answer in some sense to our rigidity problem has been given, but also the dependence between normalizers, automorphisms and especially almost inner automorphisms, has been clarified.
References:
[1] Baer, R.: Primary Abelian groups and their automorphisms. Am. J. Math. 59 (1937), 99-117. DOI 10.2307/2371564 | MR 1507222 | Zbl 0016.01404
[2] Besson, G., Courtois, G., Gallot, S.: Entropy and rigidity of locally symmetric spaces of strictly negative curvature. Geom. Func. Anal. 5 (1995), 731-799 French. DOI 10.1007/BF01897050 | MR 1354289 | Zbl 0851.53032
[3] Besson, G., Courtois, G., Gallot, S.: Minimal entropy and Mostow's rigidity theorems. Ergodic Theory Dyn. Syst. 16 (1996), 623-649. DOI 10.1017/S0143385700009019 | MR 1406425 | Zbl 0887.58030
[4] Châtelet, A.: Les groupes abéliens finis et les modules de points entiers. Bibliothèque universitaire. Travaux et mémoires de l'Université de Lille. Nouv. série II, 3. Gauthier-Villars, Lille (1925), French \99999JFM99999 51.0115.02.
[5] Croke, C.: Rigidity for surfaces of non-positive curvature. Comment. Math. Helvet. 65 (1990), 150-169. DOI 10.1007/BF02566599 | MR 1036134 | Zbl 0704.53035
[6] Croke, C., Eberlein, P., Kleiner, B.: Conjugacy and rigidity for nonpositively curved manifolds of higher rank. Topology 35 (1996), 273-286. DOI 10.1016/0040-9383(95)00031-3 | MR 1380497 | Zbl 0859.53024
[7] Duistermaat, J. J., Kolk, J. A. C.: Lie Groups. Universitext. Springer, Berlin (2000). DOI 10.1007/978-3-642-56936-4 | MR 1738431 | Zbl 0955.22001
[8] Eberlein, P.: Geometry of two-step nilpotent groups with a left invariant metric. Ann. Sci. Éc. Norm. Supér. (4) 27 (1994), 611-660. DOI 10.24033/asens.1702 | MR 1296558 | Zbl 0820.53047
[9] Fana{ï}, H.-R., Hasan-Zadeh, A.: A symplectic rigidity problem for $2$-step nilmanifolds. Houston J. Math. 43 (2017), 363-374. MR 3690121 | Zbl 06833112
[10] Gabai, D.: On the geometric and topological rigidity of the hyperbolic 3-manifolds. Bull. Am. Math. Soc., New Ser. 31 (1994), 228-232. DOI 10.1090/S0273-0979-1994-00523-3 | MR 1261238 | Zbl 0817.57015
[11] Gordon, C. S., Mao, Y.: Geodesic conjugacies of two-step nilmanifolds. Mich. Math. J. 45 (1998), 451-481. DOI 10.1307/mmj/1030132293 | MR 1653247 | Zbl 0976.53090
[12] Gordon, C. S., Mao, Y., Schueth, D.: Symplectic rigidity of geodesic flows on two-step nilmanifolds. Ann. Sci. Éc. Norm. Supér. (4) 30 (1998), 417-427. DOI 10.1016/S0012-9593(97)89927-2 | MR 1456241 | Zbl 0897.53033
[13] Gordon, C. S., Wilson, E. N.: Isospectral deformations of compact solvmanifolds. J. Differ. Geom. 19 (1984), 241-256. DOI 10.4310/jdg/1214438431 | MR 0739790 | Zbl 0523.58043
[14] Hallett, J. T., Hirsch, K. A.: Torsion-free groups having finite automorphism groups. J. Algebra 2 (1965), 287-298. DOI 10.1016/0021-8693(65)90010-4 | MR 0183789 | Zbl 0132.27301
[15] Hallett, J. T., Hirsch, K. A.: Die Konstruktion von Gruppen mit vorgeschriebenen Automorphismengruppen. J. Reine Angew. Math. 239-240 (1969), 32-46. DOI 10.1515/crll.1969.239-240.32 | MR 0257205 | Zbl 0186.03901
[16] Hulpke, A.: Normalizer calculation using automorphisms. Computational Group Theory and the Theory of Groups. AMS special session On Computational Group Theory, Davidson, USA, 2007 Contemporary Mathematics 470. AMS, Providence (2008), 105-114 L.-C. Kappe et al. MR 2478417 | Zbl 1184.20002
[17] Mackenzie, K. C. H.: General Theory of Lie Groupoids and Lie Algebroids. London Mathematical Society Lecture Note Series 213. Cambridge University Press, Cambridge (2005). DOI 10.1017/CBO9781107325883 | MR 2157566 | Zbl 1078.58011
[18] Mann, K.: Homomorphisms between diffeomorphism groups. Avaible at https://arxiv.org/abs/1206.1196v1 (2012). MR 3294298
[19] Mostow, G. D.: Strong Rigidity of Locally Symmetric Spaces. Annals of Mathematics Studies. No. 78. Princeton University Press, Princeton (1973). DOI 10.1515/9781400881833 | MR 0385004 | Zbl 0265.53039
[20] O'Neill, B.: Semi Riemannian Geometry. With Applications to Relativity. Pure and Applied Mathematics 103. Academic Press, London (1983). MR 0719023 | Zbl 0531.53051
[21] Otal, J. P.: The spectrum marked by lengths of surfaces with negative curvature. Ann. Math. (2) 131 (1990), 151-162 French. DOI 10.2307/1971511 | MR 1038361 | Zbl 0699.58018
[22] Raghunathan, M. S.: Discrete Subgroups of Lie Groups. Ergebnisse der Mathematik und ihrer Grenzgebiete. Band 68. Springer, Berlin (1972). MR 0507234 | Zbl 0254.22005
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