ENCOUNTER with MATHEMATICS

「Lie 群の離散部分群の剛性理論」

参考文献一覧

各講演に対する参考文献

金井（第１話） – Mostow の剛性定理

[Kan88] Masahiko Kanai, Tensorial ergodicity of geodesic flows, Geometry and analysis on manifolds (Katata/Kyoto, 1987), Springer, Berlin, 1988, pp. 142–157.

[Mos66] G. D. Mostow, On the conjugacy of subgroups of semisimple groups, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), Amer. Math. Soc., Providence, R.I., 1966, pp. 413–419.

[Mos68] G. D. Mostow, Quasi-conformal mappings in n-space and the rigidity of hyperbolic space forms, Inst. Hautes Études Sci. Publ. Math. No. 34 (1968), 53–104.

[Mos73] G. D. Mostow, Strong rigidity of locally symmetric spaces, Princeton University Press, Princeton, N.J., 1973, Annals of Mathematics Studies, No. 78.

[VS93] È. B. Vinberg and O. V. Shvartsman, Discrete groups of motions of spaces of constant curvature, Geometry, II, Springer, Berlin, 1993, pp. 139– 248.

納谷 – Weil の局所剛性定理

[Cal61] Eugenio Calabi, On compact, Riemannian manifolds with constant curvature. I, Proc. Sympos. Pure Math., Vol. III, American Mathematical Society, Providence, R.I., 1961, pp. 155–180.

[CV60] Eugenio Calabi and Edoardo Vesentini, On compact, locally symmetric Kähler manifolds, Ann. of Math. (2) 71 (1960), 472–507.

[Kos68] J.-L. Koszul, Formes harmoniques vectorielles sur les espaces localement symétriques, Geometry of Homogeneous Bounded Domains (C.I.M.E., 3 Ciclo, Urbino, 1967), Edizioni Cremonese, Rome, 1968, pp. 199–261.

[MM63] 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.

[Rag72] M. S. Raghunathan, Discrete subgroups of Lie groups, Springer-Verlag, New York, 1972, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68.

[Sel60] 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.

[Wei60] André Weil, On discrete subgroups of Lie groups, Ann. of Math. (2) 72 (1960), 369–384.

[Wei62] André Weil, On discrete subgroups of Lie groups. II, Ann. of Math. (2) 75 (1962), 578–602.

[Wei64] André Weil, Remarks on the cohomology of groups, Ann. of Math. (2) 80 (1964), 149–157.

金井（第 2 話） – Margulis の超剛性と調和写像

[Cor92] Kevin Corlette, Archimedean superrigidity and hyperbolic geometry, Ann. of Math. (2) 135 (1992), no. 1, 165–182.

[GS92] Mikhail Gromov and Richard Schoen, Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one, Inst. Hautes Études Sci. Publ. Math. (1992), no. 76, 165–246.

[JY86] J. Jost and S.-T. Yau, The strong rigidity of locally symmetric complex manifolds of rank one and finite volume, Math. Ann. 275 (1986), no. 2, 291–304.

[KN62a] Soji Kaneyuki and Tadashi Nagano, On certain quadratic forms related to symmetric riemannian spaces, Osaka Math. J. 14 (1962), 241–252.

[KN62b] Soji Kaneyuki and Tadashi Nagano, On the first Betti numbers of compact quotient spaces of complex semi-simple Lie groups by discrete subgroups, Sci. Papers Coll. Gen. Ed. Univ. Tokyo 12 (1962), 1–11.

[KS93] Nicholas J. Korevaar and Richard M. Schoen, Sobolev spaces and harmonic maps for metric space targets, Comm. Anal. Geom. 1 (1993), no. 3-4, 561–659.

[Mat62] Yozô Matsushima, On the first Betti number of compact quotient spaces of higher-dimensional symmetric spaces, Ann. of Math. (2) 75 (1962), 312–330.

[MSY93] Ngaiming Mok, Yum Tong Siu, and Sai-Kee Yeung, Geometric superrigidity, Invent. Math. 113 (1993), no. 1, 57–83.

[Rag95] M. S. Raghunathan, The first Betti number of compact locally symmetric spaces, Current trends in mathematics and physics, Narosa, New Delhi, 1995, pp. 116–137.

[Pan98] Pierre Pansu, Formules de Matsushima, de Garland et propriété (T) pour des groupes agissant sur des espaces symétriques ou des immeubles, Bull. Soc. Math. France 126 (1998), no. 1, 107–139.

[Siu80] Yum Tong Siu, The complex-analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds, Ann. of Math. (2) 112 (1980), no. 1, 73–111.

[YB53] K. Yano and S. Bochner, Curvature and Betti numbers, Princeton University Press, Princeton, N. J., 1953, Annals of Mathematics Studies, No. 32.

井関 – Besson–Courtois–Gallot の剛性定理

[BCG96] Gérard Besson, Gilles Courtois, and Sylvestre Gallot, Minimal entropy and Mostow’s rigidity theorems, Ergodic Theory Dynam. Systems 16 (1996), no. 4, 623–649

[BCG98] G. Besson, G. Courtois, and S. Gallot, A real Schwarz lemma and some applications, Rend. Mat. Appl. (7) 18 (1998), no. 2, 381–410.

[Bou96] Marc Bourdon, Sur le birapport au bord des cat(−1)-espaces, Inst. Hautes Études Sci. Publ. Math. (1996), no. 83, 95–104.

[DE86] Adrien Douady and Clifford J. Earle, Conformally natural extension of homeomorphisms of the circle, Acta Math. 157 (1986), no. 1-2, 23–48.

[Ham90] Ursula Hamenstädt, Entropy-rigidity of locally symmetric spaces of negative curvature, Ann. of Math. (2) 131 (1990), no. 1, 35–51.

[Pan89] Pierre Pansu, Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math. (2) 129 (1989), no. 1, 1–60.

[Pan97] Pierre Pansu, Volume, courbure et entropie (d’après G. Besson, G. Courtois et S. Gallot), Astérisque (1997), no. 245, Exp. No. 823, 3, 83–103, Séminaire Bourbaki, Vol. 1996/97.

金井（第 3 話） – 群作用の局所剛性・非存在定理

[Ben96] Elie Jerom Benveniste, Rigidity and deformations of lattice actions preserving geometric structures, Ph.D. thesis, Chicago University, 1996.

[BM99a] M. Burger and N. Monod, Bounded cohomology of lattices in higher rank Lie groups, J. Eur. Math. Soc. (JEMS) 1 (1999), no. 2, 199–235.

[BM99b] M. Burger and N. Monod, Erratum: “Bounded cohomology of lattices in higher rank Lie groups” [J. Eur. Math. Soc. (JEMS) 1 (1999), no. 2, 199–235; 1694584], J. Eur. Math. Soc. (JEMS) 1 (1999), no. 3, 338.

[FSa] Benson Farb and Peter Shalen, Groups of real-analytic diffeomorphisms of the circle, Preprint.

[FSb] Benson Farb and Peter Shalen, Lattice actions, 3-manifolds, and homology, Preprint.

[FS99] Benson Farb and Peter Shalen, Real-analytic actions of lattices, Invent. Math. 135 (1999), no. 2, 273–296.

[Ghy92] Étienne Ghys, Déformations de flots d’Anosov et de groupes fuchsiens, Ann. Inst. Fourier (Grenoble) 42 (1992), no. 1-2, 209–247.

[Ghy93] Étienne Ghys, Rigidité différentiable des groupes fuchsiens, Inst. Hautes Études Sci. Publ. Math. (1993), no. 78, 163–185 (1994).

[Ghy99] Étienne Ghys, Actions de réseaux sur le cercle, Invent. Math. 137 (1999), no. 1, 199–231.

[Hur92] Steven Hurder, Rigidity for Anosov actions of higher rank lattices, Ann. of Math. (2) 135 (1992), no. 2, 361–410.

[Hur94] Steven Hurder, A survey of rigidity theory for Anosov actions, Differential topology, foliations, and group actions (Rio de Janeiro, 1992), Amer. Math. Soc., Providence, RI, 1994, pp. 143–173.

[Kan] Masahiko Kanai, A remark on local rigidity of conformal actions on the

sphere, to appear in Math. Research Lett.

[Kan96] M. Kanai, A new approach to the rigidity of discrete group actions, Geom. Funct. Anal. 6 (1996), no. 6, 943–1056.

[KL91] A. Katok and J. Lewis, Local rigidity for certain groups of toral automorphisms, Israel J. Math. 75 (1991), no. 2-3, 203–241.

[KLZ96] A. Katok, J. Lewis, and R. Zimmer, Cocycle superrigidity and rigidity

for lattice actions on tori, Topology 35 (1996), no. 1, 27–38.

[KS] N. J. Korevaar and R. Schoen, Global existence theorems for harmonic maps: Finite rank spaces and an approach to rigidity for smooth actions, Preprint.

[KS96] A. Katok and R. J. Spatzier, Nonstationary normal forms and rigidity of group actions, Electron. Res. Announc. Amer. Math. Soc. 2 (1996), no. 3, 124–133 (electronic).

[KS97] A. Katok and R. J. Spatzier, Differential rigidity of Anosov actions of higher rank abelian groups and algebraic lattice actions, Tr. Mat. Inst. Steklova 216 (1997), no. Din. Sist. i Smezhnye Vopr., 292–319.

[KY97] A. Kononenko and C. B. Yue, Cohomology and rigidity of Fuchsian

groups, Israel J. Math. 97 (1997), 51–59.

[Lab] François Labourie, Large groups actions on manifolds, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), vol. 1998, pp. 371–380 (electronic).

[Lew91] James W. Lewis, Infinitesimal rigidity for the action of SL(n, ℤ) on 𝕋^{n},

Trans. Amer. Math. Soc. 324 (1991), no. 1, 421–445.

[MQ] Gregory A. Margulis and Nantian Qian, Rigidity of weakly hyperbolic

actions of higher real rank semisimple lie groups and their lattices.

[Wei97] Shmuel Weinberger, SL(n, ℤ) cannot act on small tori, Geometric topology (Athens, GA, 1993), Amer. Math. Soc., Providence, RI, 1997, pp. 406–408.

[Wit94] Dave Witte, Arithmetic groups of higher ℚ-rank cannot act on 1-manifolds, Proc. Amer. Math. Soc. 122 (1994), no. 2, 333–340.

[Zeg99] Abdelghani Zeghib, Quelques remarques sur les actions analytiques des réseaux des groupes de Lie de rang supérieur, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), no. 9, 799–804.

[Zim90] Robert J. Zimmer, Infinitesimal rigidity for smooth actions of discrete subgroups of Lie groups, J. Differential Geom. 31 (1990), no. 2, 301– 322.

その他

一般的解説

[GP91] M. Gromov and P. Pansu, Rigidity of lattices: an introduction, Geometric topology: recent developments (Montecatini Terme, 1990), Springer, Berlin, 1991, pp. 39–137.

[Pan95] Pierre Pansu, Sous-groupes discrets des groupes de Lie: rigidité, arithméticité, Astérisque (1995), no. 227, Exp. No. 778, 3, 69–105, Séminaire Bourbaki, Vol. 1993/94.

[Spa95] R. J. Spatzier, Harmonic analysis in rigidity theory, Ergodic theory and its connections with harmonic analysis (Alexandria, 1993), Cambridge Univ. Press, Cambridge, 1995, pp. 153–205.

教科書・モノグラフ [BGS85] Werner Ballmann, Mikhael Gromov, and Viktor Schroeder, Manifolds

of nonpositive curvature, Birkhäuser Boston Inc., Boston, Mass., 1985.

[BP92] Riccardo Benedetti and Carlo Petronio, Lectures on hyperbolic geometry, Springer-Verlag, Berlin, 1992.

[Ebe96] Patrick B. Eberlein, Geometry of nonpositively curved manifolds, University of Chicago Press, Chicago, IL, 1996.

[Fer98] Renato Feres, Dynamical systems and semisimple groups: an introduction, Cambridge University Press, Cambridge, 1998.

[Jos97] Jürgen Jost, Nonpositive curvature: geometric and analytic aspects, Birkhäuser Verlag, Basel, 1997.

[Mar91] G. A. Margulis, Discrete subgroups of semisimple Lie groups, Springer-Verlag, Berlin, 1991.

[Mos73] G. D. Mostow, Strong rigidity of locally symmetric spaces, Princeton University Press, Princeton, N.J., 1973, Annals of Mathematics Studies, No. 78.

[Rag72] M. S. Raghunathan, Discrete subgroups of Lie groups, Springer-Verlag, New York, 1972, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68.

[Rat94] John G. Ratcliffe, Foundations of hyperbolic manifolds, Springer-Verlag, New York, 1994.

[VS93] È. B. Vinberg and O. V. Shvartsman, Discrete groups of motions of spaces of constant curvature, Geometry, II, Springer, Berlin, 1993, pp. 139– 248.

[Zim84] Robert J. Zimmer, Ergodic theory and semisimple groups, Birkhäuser Verlag, Basel, 1984.

日本語で書かれた解説記事等

[いぜ] 井関裕靖，「G. Besson, G. Courtois and S. Gallot による Mostow の剛

性定理の新証明」，数学，第 49 巻， 200–211. [い 1] 伊原信一郎，「リー群の離散部分群」，上智大学数学講究録， no. 16, 1984. [い 2] 伊原信一郎，「G. A. Margulis 氏の業績」，数学，第 31 巻， 43–50. [か 1] 金井雅彦，「力学系の不変幾何構造と剛性問題」，数学，第 52 巻， 43–52. [か 2] 金井雅彦，「群作用の剛性問題」，数学の楽しみ第 18 号に掲載の予定． [さ] 佐竹一郎，「リー群の話」に収録の付録「G. A. マルグリス」，日本評論社．