Geodesic
An application of Lie groupoids to a rigidity problem of 2-step nilmanifolds
Mathematica Bohemica, Tome 144 (2019) no. 2, pp. 149-160
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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.
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.
DOI : 10.21136/MB.2018.0041-17
Classification : 22A22, 22F05, 53C24
Keywords: nilpotent Lie group; isometric nilmanifolds; normalizer; Lie algebroid; normal subgroupoid system; inner automorphism 
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Fanaï, Hamid-Reza; Hasan-Zadeh, Atefeh. An application of Lie groupoids to a rigidity problem of 2-step nilmanifolds. Mathematica Bohemica, Tome 144 (2019) no. 2, pp. 149-160. doi: 10.21136/MB.2018.0041-17

[1] Baer, R.: Primary Abelian groups and their automorphisms. Am. J. Math. 59 (1937), 99-117. | DOI | MR | JFM

[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 | MR | JFM

[3] Besson, G., Courtois, G., Gallot, S.: Minimal entropy and Mostow's rigidity theorems. Ergodic Theory Dyn. Syst. 16 (1996), 623-649. | DOI | MR | JFM

[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 | MR | JFM

[6] Croke, C., Eberlein, P., Kleiner, B.: Conjugacy and rigidity for nonpositively curved manifolds of higher rank. Topology 35 (1996), 273-286. | DOI | MR | JFM

[7] Duistermaat, J. J., Kolk, J. A. C.: Lie Groups. Universitext. Springer, Berlin (2000). | DOI | MR | JFM

[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 | MR | JFM

[9] Fana{ï}, H.-R., Hasan-Zadeh, A.: A symplectic rigidity problem for $2$-step nilmanifolds. Houston J. Math. 43 (2017), 363-374. | MR | JFM

[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 | MR | JFM

[11] Gordon, C. S., Mao, Y.: Geodesic conjugacies of two-step nilmanifolds. Mich. Math. J. 45 (1998), 451-481. | DOI | MR | JFM

[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 | MR | JFM

[13] Gordon, C. S., Wilson, E. N.: Isospectral deformations of compact solvmanifolds. J. Differ. Geom. 19 (1984), 241-256. | DOI | MR | JFM

[14] Hallett, J. T., Hirsch, K. A.: Torsion-free groups having finite automorphism groups. J. Algebra 2 (1965), 287-298. | DOI | MR | JFM

[15] Hallett, J. T., Hirsch, K. A.: Die Konstruktion von Gruppen mit vorgeschriebenen Automorphismengruppen. J. Reine Angew. Math. 239-240 (1969), 32-46. | DOI | MR | JFM

[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 | JFM

[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 | MR | JFM

[18] Mann, K.: Homomorphisms between diffeomorphism groups. Avaible at (2012). | arXiv | MR

[19] Mostow, G. D.: Strong Rigidity of Locally Symmetric Spaces. Annals of Mathematics Studies. No. 78. Princeton University Press, Princeton (1973). | DOI | MR | JFM

[20] O'Neill, B.: Semi Riemannian Geometry. With Applications to Relativity. Pure and Applied Mathematics 103. Academic Press, London (1983). | MR | JFM

[21] Otal, J. P.: The spectrum marked by lengths of surfaces with negative curvature. Ann. Math. (2) 131 (1990), 151-162 French. | DOI | MR | JFM

[22] Raghunathan, M. S.: Discrete Subgroups of Lie Groups. Ergebnisse der Mathematik und ihrer Grenzgebiete. Band 68. Springer, Berlin (1972). | MR | JFM

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