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Talia Fernós
Relative property (T) and linear groups
(La propriété (T) relative et les groupes linéaires)
Annales de l'institut Fourier, 56 no. 6 (2006), p. 1767-1804
Article: subscription required | Reviews MR 2282675 | Zbl 1175.22004
Class. Math.: 20F99, 20E22, 20G25, 46G99
Keywords: Relative property (T), group extensions, linear algebraic groups
[1] H. Bass, J. Milnor & J.-P. Serre, “Solution of the congruence subgroup problem for ${\rm SL}_{n}\,(n\ge 3)$ and ${\rm Sp}_{2n}\,(n\ge 2)$”, Inst. Hautes Études Sci. Publ. Math. (1967), p. 59-137
Numdam | MR 244257 | Zbl 0174.05203
[2] Hyman Bass, “Groups of integral representation type”, Pacific Journal of Math. 86 (1980), p. 15-51
Article | MR 586867 | Zbl 0444.20006
[3] A. Borel & J. Tits, “Groupes réductifs”, Publ. Math. IHÉS 27 (1965), p. 55-150
Numdam | MR 207712 | Zbl 0145.17402
[4] Armand Borel, Linear algebraic groups, Springer-Verlag, 1991 MR 1102012 | Zbl 0726.20030
[5] Marc Burger, “Kazhdan constants for $\textnormal{SL}_{3}(\mathbb{Z})$”, J. reine angew. Math. 413 (1991), p. 36-67 MR 1089795 | Zbl 0704.22009
[6] Damien Gaboriau & Sorin Popa, “An uncountable family of non-orbit equivalen actions of $F_n$”, arXiv:math.GR/0306011, 2003
arXiv
[7] Larry Joel Goldstein, Analytic number theory, Prentice-Hall, 1971 MR 498335 | Zbl 0226.12001
[8] Pierre de la Harpe & Alain Valette, “La propriété (T) de Kazhdan pour les groupes localement compacts”, Asterisque 175 (1989), p. 1-157 Zbl 0759.22001
[9] G. Hochschild, The structure of Lie groups, Holden-Day Inc., 1965 MR 207883 | Zbl 0131.02702
[10] James E. Humphreys, Linear Algebraic Groups, Springer, 1998 MR 396773 | Zbl 0471.20029
[11] P. Jolissaint, “Borel cocycles, approximation properties and relative property (T)”, Ergod. Th. & Dynam. Sys. 20 (2000), p. 483-499 MR 1756981 | Zbl 0955.22008
[12] Martin Kassabov & Nikolay Nikolov, “Universal lattices and property $\tau $”, http://arxiv.org/ pdf/math.GR/0502112, 2004
arXiv | Zbl 05052841
[13] D. Kazhdan, “Connection of the dual space of a group with the structure of its closed subgroups”, Functional Analysis and its Applications 1 (1967), p. 63-65 MR 209390 | Zbl 0168.27602
[14] Alexander Lubotzky, Shahar Mozes & M. S. Raghunathan, “The word and Riemannian metrics on lattices of semisimple groups”, Inst. Hautes Études Sci. Publ. Math. (2000), p. 5-53 (2001)
Numdam | MR 1828742 | Zbl 0988.22007
[15] George W. Mackey, “A theorem of Stone and von Neumann”, Duke Math. J. 16 (1949), p. 313-326
Article | MR 30532 | Zbl 0036.07703
[16] George W. Mackey, “Induced representations of locally compact groups, I”, Ann. of Math. (2) 55 (1952), p. 101-139 MR 44536 | Zbl 0046.11601
[17] G. A. Margulis, Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] 17, Springer-Verlag, 1991 MR 1090825 | Zbl 0732.22008
[18] G.A. Margulis, “Explicit constructions of concentrators”, Problems Information Transmission 9 (1973), p. 325-332 MR 484767 | Zbl 0312.22011
[19] Andrés Navas, “Quelques nouveaux phénomènes de rang 1 pour les groupes de difféomorphismes du cercle”, Comment. Math. Helv. 80 (2005), p. 355-375 MR 2142246 | Zbl 1080.57002
[20] Sorin Popa, “On a class of type II$_1$ factors with Betti numbers invariants”, MSRI preprint no. 2001-0024 math.OA/0209310, 2003
arXiv | MR 1867564
[21] Sorin Popa, “Strong rigidity of II$_1$ factors arising from malleable actions of w-rigid groups, Part I”, arXiv:math.OA/0305306, 2003
arXiv
[22] Sorin Popa, “Some computations of 1-cohomology groups and constructions of non-orbit equivalent actions”, arXiv:math.OA/0407199, 2004
arXiv
[23] Y. Shalom, “Measurable group theory”, Preprint, 2005 MR 2185757 | Zbl 02212149
[24] Yehuda Shalom, “Bounded generation and Kazhdan’s property (T)”, Inst. Hautes Études Sci. Publ. Math. (1999), p. 145-168 (2001)
Numdam | MR 1813225 | Zbl 0980.22017
[25] T.A. Springer, Linear algebraic groups, 2nd edition, Birkhäuser, 1998 MR 1642713
[26] A. A. Suslin, “The structure of the special linear group over rings of polynomials”, Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), p. 235-252, 477 MR 472792 | Zbl 0378.13002
[27] J. Tits, Classification of algebraic semisimple groups in algebraic groups and discrete subgroups, Proc. Symp. Pure Math. IX Amer. Math. Soc., 1966 MR 224710 | Zbl 0238.20052
[28] Asger Törnquist, “Orbit equivalence and $F_n$ actions”, Preprint, 2004
[29] Alain Valette, “Group pairs with property (T), from arithmetic lattices”, Geom. Dedicata 112 (2005), p. 183-196 MR 2163898 | Zbl 1076.22012
[30] P. S. Wang, “On isolated points in the dual spaces of locally compact groups”, Mathematische Annalen 218 (1975), p. 19-34
Article | MR 384993 | Zbl 0332.22009
[31] H. Whitney, “Elementary structure of real algebraic varieties”, The Annals of Mathematics 66 (1957), p. 545-556 MR 95844 | Zbl 0078.13403
[32] R.J. Zimmer, Ergodic theory and semisimple groups, Birkhäuser, 1984 MR 776417 | Zbl 0571.58015
Talia Fernós
Relative property (T) and linear groups
(La propriété (T) relative et les groupes linéaires)
Annales de l'institut Fourier, 56 no. 6 (2006), p. 1767-1804
Article: subscription required | Reviews MR 2282675 | Zbl 1175.22004
Class. Math.: 20F99, 20E22, 20G25, 46G99
Keywords: Relative property (T), group extensions, linear algebraic groups
Résumé - Abstract
Relative property (T) has recently been used to show the existence of a variety of new rigidity phenomena, for example in von Neumann algebras and the study of orbit-equivalence relations. However, until recently there were few examples of group pairs with relative property (T) available through the literature. This motivated the following result: A finitely generated group $\Gamma $ admits a special linear representation with non-amenable $R$-Zariski closure if and only if it acts on an Abelian group $A$ (of finite nonzero $Q$-rank) so that the corresponding group pair $(\Gamma \ltimesA,A)$ has relative property (T).
The proof is constructive. The main ingredients are Furstenberg’s celebrated lemma about invariant measures on projective spaces and the spectral theorem for the decomposition of unitary representations of Abelian groups. Methods from algebraic group theory, such as the restriction of scalars functor, are also employed.
Bibliography
Numdam | MR 244257 | Zbl 0174.05203
[2] Hyman Bass, “Groups of integral representation type”, Pacific Journal of Math. 86 (1980), p. 15-51
Article | MR 586867 | Zbl 0444.20006
[3] A. Borel & J. Tits, “Groupes réductifs”, Publ. Math. IHÉS 27 (1965), p. 55-150
Numdam | MR 207712 | Zbl 0145.17402
[4] Armand Borel, Linear algebraic groups, Springer-Verlag, 1991 MR 1102012 | Zbl 0726.20030
[5] Marc Burger, “Kazhdan constants for $\textnormal{SL}_{3}(\mathbb{Z})$”, J. reine angew. Math. 413 (1991), p. 36-67 MR 1089795 | Zbl 0704.22009
[6] Damien Gaboriau & Sorin Popa, “An uncountable family of non-orbit equivalen actions of $F_n$”, arXiv:math.GR/0306011, 2003
arXiv
[7] Larry Joel Goldstein, Analytic number theory, Prentice-Hall, 1971 MR 498335 | Zbl 0226.12001
[8] Pierre de la Harpe & Alain Valette, “La propriété (T) de Kazhdan pour les groupes localement compacts”, Asterisque 175 (1989), p. 1-157 Zbl 0759.22001
[9] G. Hochschild, The structure of Lie groups, Holden-Day Inc., 1965 MR 207883 | Zbl 0131.02702
[10] James E. Humphreys, Linear Algebraic Groups, Springer, 1998 MR 396773 | Zbl 0471.20029
[11] P. Jolissaint, “Borel cocycles, approximation properties and relative property (T)”, Ergod. Th. & Dynam. Sys. 20 (2000), p. 483-499 MR 1756981 | Zbl 0955.22008
[12] Martin Kassabov & Nikolay Nikolov, “Universal lattices and property $\tau $”, http://arxiv.org/ pdf/math.GR/0502112, 2004
arXiv | Zbl 05052841
[13] D. Kazhdan, “Connection of the dual space of a group with the structure of its closed subgroups”, Functional Analysis and its Applications 1 (1967), p. 63-65 MR 209390 | Zbl 0168.27602
[14] Alexander Lubotzky, Shahar Mozes & M. S. Raghunathan, “The word and Riemannian metrics on lattices of semisimple groups”, Inst. Hautes Études Sci. Publ. Math. (2000), p. 5-53 (2001)
Numdam | MR 1828742 | Zbl 0988.22007
[15] George W. Mackey, “A theorem of Stone and von Neumann”, Duke Math. J. 16 (1949), p. 313-326
Article | MR 30532 | Zbl 0036.07703
[16] George W. Mackey, “Induced representations of locally compact groups, I”, Ann. of Math. (2) 55 (1952), p. 101-139 MR 44536 | Zbl 0046.11601
[17] G. A. Margulis, Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] 17, Springer-Verlag, 1991 MR 1090825 | Zbl 0732.22008
[18] G.A. Margulis, “Explicit constructions of concentrators”, Problems Information Transmission 9 (1973), p. 325-332 MR 484767 | Zbl 0312.22011
[19] Andrés Navas, “Quelques nouveaux phénomènes de rang 1 pour les groupes de difféomorphismes du cercle”, Comment. Math. Helv. 80 (2005), p. 355-375 MR 2142246 | Zbl 1080.57002
[20] Sorin Popa, “On a class of type II$_1$ factors with Betti numbers invariants”, MSRI preprint no. 2001-0024 math.OA/0209310, 2003
arXiv | MR 1867564
[21] Sorin Popa, “Strong rigidity of II$_1$ factors arising from malleable actions of w-rigid groups, Part I”, arXiv:math.OA/0305306, 2003
arXiv
[22] Sorin Popa, “Some computations of 1-cohomology groups and constructions of non-orbit equivalent actions”, arXiv:math.OA/0407199, 2004
arXiv
[23] Y. Shalom, “Measurable group theory”, Preprint, 2005 MR 2185757 | Zbl 02212149
[24] Yehuda Shalom, “Bounded generation and Kazhdan’s property (T)”, Inst. Hautes Études Sci. Publ. Math. (1999), p. 145-168 (2001)
Numdam | MR 1813225 | Zbl 0980.22017
[25] T.A. Springer, Linear algebraic groups, 2nd edition, Birkhäuser, 1998 MR 1642713
[26] A. A. Suslin, “The structure of the special linear group over rings of polynomials”, Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), p. 235-252, 477 MR 472792 | Zbl 0378.13002
[27] J. Tits, Classification of algebraic semisimple groups in algebraic groups and discrete subgroups, Proc. Symp. Pure Math. IX Amer. Math. Soc., 1966 MR 224710 | Zbl 0238.20052
[28] Asger Törnquist, “Orbit equivalence and $F_n$ actions”, Preprint, 2004
[29] Alain Valette, “Group pairs with property (T), from arithmetic lattices”, Geom. Dedicata 112 (2005), p. 183-196 MR 2163898 | Zbl 1076.22012
[30] P. S. Wang, “On isolated points in the dual spaces of locally compact groups”, Mathematische Annalen 218 (1975), p. 19-34
Article | MR 384993 | Zbl 0332.22009
[31] H. Whitney, “Elementary structure of real algebraic varieties”, The Annals of Mathematics 66 (1957), p. 545-556 MR 95844 | Zbl 0078.13403
[32] R.J. Zimmer, Ergodic theory and semisimple groups, Birkhäuser, 1984 MR 776417 | Zbl 0571.58015
© Annales de L'Institut Fourier - ISSN (électronique) : 1777-5310