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 ${\rm {c}{a}{t}}(-1)$-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 ${\rm {S}{L}}(n,{\mathbb{Z} })$ on ${\mathbb{T} }\sp n$, 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, ${\rm {S}{L}}(n,{\mathbb{Z} })$ 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 ${\mathbb{Q} }$-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．