Character rigidity for special linear groups. (English) Zbl 1347.20051
In the paper the characters of the group \(\mathrm{SL}_n(R)\) where \(R\) is an infinite field or a localization of an order in a number field are studied.
The main results of the paper are the following. For the group \(\mathrm{SL}_2(k)\) where \(k\) is an infinite field a complete classification of characters is given. Moreover, the authors prove a character rigidity result for groups \(\mathrm{SL}_2(B^{-1}S)\) where \(B\) is an order in a number field \(K\), \(S\subset B\) is a multiplicative subset, \(B^{-1}S\) denotes the localization. It is assumed that \(B^{-1}S\) contains an infinite number of units.
The authors show that character rigidity implies a non-existence result for invariant random subgroups. This was also shown independently by A. Dudko and K. Medynets [Groups Geom. Dyn. 8, No. 2, 375-389 (2014; Zbl 1328.20012)]. As a particular consequence it is proved the non-existence of invariant random subgroups of \(\mathrm{SL}_n(k)\) for \(n\geq 2\). It is proved that for any infinite field, any non-trivial measure-preserving ergodic action of \(\mathrm{PSL}_2(k)\) on a probability measure space is essentially free. Similar results are derived for groups \(\mathrm{PSL}_n(k)\) and \(\mathrm{PSL}_n(B^{-1}S)\), \(n\geq 3\) and \(\mathrm{PSL}_2(B^{-1}S)\) in the case when \(B^{-1}S\) contains an infinite number of units.
The main results of the paper are the following. For the group \(\mathrm{SL}_2(k)\) where \(k\) is an infinite field a complete classification of characters is given. Moreover, the authors prove a character rigidity result for groups \(\mathrm{SL}_2(B^{-1}S)\) where \(B\) is an order in a number field \(K\), \(S\subset B\) is a multiplicative subset, \(B^{-1}S\) denotes the localization. It is assumed that \(B^{-1}S\) contains an infinite number of units.
The authors show that character rigidity implies a non-existence result for invariant random subgroups. This was also shown independently by A. Dudko and K. Medynets [Groups Geom. Dyn. 8, No. 2, 375-389 (2014; Zbl 1328.20012)]. As a particular consequence it is proved the non-existence of invariant random subgroups of \(\mathrm{SL}_n(k)\) for \(n\geq 2\). It is proved that for any infinite field, any non-trivial measure-preserving ergodic action of \(\mathrm{PSL}_2(k)\) on a probability measure space is essentially free. Similar results are derived for groups \(\mathrm{PSL}_n(k)\) and \(\mathrm{PSL}_n(B^{-1}S)\), \(n\geq 3\) and \(\mathrm{PSL}_2(B^{-1}S)\) in the case when \(B^{-1}S\) contains an infinite number of units.
Reviewer: Dmitry Artamonov (Moskva)
MSC:
| 20G05 | Representation theory for linear algebraic groups |
| 20G15 | Linear algebraic groups over arbitrary fields |
| 20G30 | Linear algebraic groups over global fields and their integers |
| 22D10 | Unitary representations of locally compact groups |
| 22F10 | Measurable group actions |
| 43A35 | Positive definite functions on groups, semigroups, etc. |
| 37A05 | Dynamical aspects of measure-preserving transformations |
Citations:
Zbl 1328.20012References:
| [1] | Abért M., Avni N. and Wilson J., A strong simplicity property for projective special linear groups, personal communication.; Abért, M.; Avni, N.; Wilson, J., A strong simplicity property for projective special linear groups |
| [2] | Abért M., Glasner Y. and Virag B., Kesten’s theorem for invariant random subgroups, preprint 2012, . <pub-id pub-id-type=”ThomsonISI“>http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000331508000001&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=b7bc2757938ac7a7a821505f8243d9f3; Abért, M.; Glasner, Y.; Virag, B., Kesten’s theorem for invariant random subgroups (2012) |
| [3] | Bekka B., Operator-algebraic superrigidity for \({\text{SL}_n(\mathbb{Z})} , {n\geq 3} \), Invent. Math. 169 (2007), no. 2, 401-425.; Bekka, B., Operator-algebraic superrigidity for \({\text{SL}_n(\mathbb{Z})} , {n\geq 3} \), Invent. Math., 169, 2, 401-425 (2007) · Zbl 1135.22009 |
| [4] | Burger M., Ozawa N. and Thom A., On Ulam stability, Israel J. Math. 193 (2013), 109-129.; Burger, M.; Ozawa, N.; Thom, A., On Ulam stability, Israel J. Math., 193, 109-129 (2013) · Zbl 1271.22003 |
| [5] | Choda M., Group factors of the Haagerup type, Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), 174-177.; Choda, M., Group factors of the Haagerup type, Proc. Japan Acad. Ser. A Math. Sci., 59, 174-177 (1983) · Zbl 0523.46038 |
| [6] | Cimpric J., Netzer T. and Marschall M., Closures of quadratic modules, Israel J. Math. 183 (2011), no. 1, 445-474.; Cimpric, J.; Netzer, T.; Marschall, M., Closures of quadratic modules, Israel J. Math., 183, 1, 445-474 (2011) · Zbl 1235.13021 |
| [7] | Connes A., Correspondences, handwritten notes 1980.; Connes, A., Correspondences (1980) |
| [8] | Connes A. and Jones V., Property T for von Neumann algebras, Bull. Lond. Math. Soc. 17 (1985), 57-62.; Connes, A.; Jones, V., Property T for von Neumann algebras, Bull. Lond. Math. Soc., 17, 57-62 (1985) · Zbl 1190.46047 |
| [9] | Creutz D. and Peterson J., Stabilizers of ergodic actions of lattices and commensurators, preprint 2012.; Creutz, D.; Peterson, J., Stabilizers of ergodic actions of lattices and commensurators (2012) · Zbl 1375.37007 |
| [10] | Dixmier J., Von Neumann algebras, North-Holland Math. Lib. 27, North-Holland, Amsterdam 1981.; Dixmier, J., Von Neumann algebras (1981) · Zbl 0473.46040 |
| [11] | Dudko A. and Medynets K., Finite factor representations of Higman-Thompson groups, Groups Geom. Dyn., to appear.; Dudko, A.; Medynets, K., Finite factor representations of Higman-Thompson groups, Groups Geom. Dyn. · Zbl 1328.20012 |
| [12] | Dye H. A., On groups of measure preserving transformations. I, Amer. J. Math. 81 (1959), no. 1, 119-159.; Dye, H. A., On groups of measure preserving transformations. I, Amer. J. Math., 81, 1, 119-159 (1959) · Zbl 0087.11501 |
| [13] | Dye H. A., On the ergodic mixing theorem, Trans. Amer. Math. Soc. 118 (1965), 123-130.; Dye, H. A., On the ergodic mixing theorem, Trans. Amer. Math. Soc., 118, 123-130 (1965) · Zbl 0142.13601 |
| [14] | Feldman J. and Moore C. C., Ergodic equivalence relations, cohomology, and von Neumann algebras. II, Trans. Amer. Math. Soc. 234 (1977), no. 2, 325-259.; Feldman, J.; Moore, C. C., Ergodic equivalence relations, cohomology, and von Neumann algebras. II, Trans. Amer. Math. Soc., 234, 2, 325-259 (1977) · Zbl 0369.22010 |
| [15] | Gamm C., ε-representations of groups and Ulam stability, Master thesis, Georg-August-Universität Göttingen, Göttingen 2011.; Gamm, C., ε-representations of groups and Ulam stability (2011) |
| [16] | Gluck D., Sharper character value estimates for groups of Lie type, J. Algebra 174 (1995), 229-266.; Gluck, D., Sharper character value estimates for groups of Lie type, J. Algebra, 174, 229-266 (1995) · Zbl 0842.20014 |
| [17] | Grunewald F. and Schwermer J., Free nonabelian quotients of \({\text{SL}_2}\) over orders of imaginary quadratic numberfields, J. Algebra 69 (1981), 298-304.; Grunewald, F.; Schwermer, J., Free nonabelian quotients of \({\text{SL}_2}\) over orders of imaginary quadratic numberfields, J. Algebra, 69, 298-304 (1981) · Zbl 0461.20026 |
| [18] | Haagerup U., An example of a non-nuclear \({C^*}\)-algebra, which has the metric approximation property, Invent. Math. 50 (1978/79), 279-293.; Haagerup, U., An example of a non-nuclear \({C^*}\)-algebra, which has the metric approximation property, Invent. Math., 50, 279-293 (197879) · Zbl 0408.46046 |
| [19] | Harish-Chandra , On the characters of a semisimple Lie group, Bull. Amer. Math. Soc. 61 (1955), no. 5, 389-396.; Harish-Chandra, On the characters of a semisimple Lie group, Bull. Amer. Math. Soc., 61, 5, 389-396 (1955) · Zbl 0065.35002 |
| [20] | Kadison R. V., A generalized Schwarz inequality and algebraic invariants for operator algebras, Ann. of Math. (2) 56 (1952), 494-503.; Kadison, R. V., A generalized Schwarz inequality and algebraic invariants for operator algebras, Ann. of Math. (2), 56, 494-503 (1952) · Zbl 0047.35703 |
| [21] | Kirillov A. A., Positive definite functions on a group of matrices with elements from a discrete field, Sov. Math. Dokl. 6 (1965), 707-709.; Kirillov, A. A., Positive definite functions on a group of matrices with elements from a discrete field, Sov. Math. Dokl., 6, 707-709 (1965) · Zbl 0133.37505 |
| [22] | Margulis G. A., Non-uniform lattices in semisimple algebraic groups, Lie groups and their representations (Budapest 1971), Wiley, New York (1975), 371-553.; Margulis, G. A., Non-uniform lattices in semisimple algebraic groups, Lie groups and their representations, 371-553 (1975) · Zbl 0316.22009 |
| [23] | Morris D. W., Bounded generation of \({\text{SL}(n,A)} \) (after D. Carter, G. Keller, and E. Paige), New York J. Math. 13 (2007), 383-421.; Morris, D. W., Bounded generation of \({\text{SL}(n,A)} \) (after D. Carter, G. Keller, and E. Paige), New York J. Math., 13, 383-421 (2007) · Zbl 1137.20045 |
| [24] | Mostow G. D., Strongrigidity of locally symmetric spaces, Ann. of Math. Stud. 78, Princeton University Press, Princeton 1973.; Mostow, G. D., Strongrigidity of locally symmetric spaces (1973) · Zbl 0265.53039 |
| [25] | von Neumann J. and Wigner E., Minimally almost periodic groups, Ann. of Math. (2) 41 (1940), 746-750.; von Neumann, J.; Wigner, E., Minimally almost periodic groups, Ann. of Math. (2), 41, 746-750 (1940) · JFM 66.0544.02 |
| [26] | Nevo A. and Zimmer R., Homogenous projection factors for the action of semi-simple Lie groups, Invent. Math. 138 (1999), 229-252.; Nevo, A.; Zimmer, R., Homogenous projection factors for the action of semi-simple Lie groups, Invent. Math., 138, 229-252 (1999) · Zbl 0936.22007 |
| [27] | Ovcinnikov S. V., Positive definite functions on Chevalley groups, Funct. Anal. Appl. 5 (1971), 79-80.; Ovcinnikov, S. V., Positive definite functions on Chevalley groups, Funct. Anal. Appl., 5, 79-80 (1971) · Zbl 0247.43015 |
| [28] | Ozawa N., Quasi-homomorphism rigidity with non-commutative targets, J. reine angew. Math. 655 (2011), 89-104.; Ozawa, N., Quasi-homomorphism rigidity with non-commutative targets, J. reine angew. Math., 655, 89-104 (2011) · Zbl 1222.22002 |
| [29] | Popa S., Orthogonal pairs of \({\ast}\)-subalgebras in finite von Neumann algebras, J. Operator Theory 9 (1983), no. 2, 253-268.; Popa, S., Orthogonal pairs of \({\ast}\)-subalgebras in finite von Neumann algebras, J. Operator Theory, 9, 2, 253-268 (1983) · Zbl 0521.46048 |
| [30] | Popa S., Correspondences, INCREST Preprint 56/1986, 1986.; Popa, S., Correspondences (1986) |
| [31] | Popa S., On a class of type II \({{}_1}\) factors with Betti numbers invariants, Ann. of Math. (2) 163 (2006), 809-899.; Popa, S., On a class of type II \({{}_1}\) factors with Betti numbers invariants, Ann. of Math. (2), 163, 809-899 (2006) · Zbl 1120.46045 |
| [32] | Prasad G., Strong rigidity of Q-rank 1 lattices, Invent. Math. 21 (1973), 255-286.; Prasad, G., Strong rigidity of Q-rank 1 lattices, Invent. Math., 21, 255-286 (1973) · Zbl 0264.22009 |
| [33] | Robertson A. G., Property (T) for II \({{}_1}\) factors and unitary representations of Kazhdan groups, Math. Ann. 296 (1993), 547-555.; Robertson, A. G., Property (T) for II \({{}_1}\) factors and unitary representations of Kazhdan groups, Math. Ann., 296, 547-555 (1993) · Zbl 0787.22008 |
| [34] | Segal I. and von Neumann J., A theorem on unitary representations of semi-simple Lie groups, Ann. of Math. (2) 52 (1950), 509-517.; Segal, I.; von Neumann, J., A theorem on unitary representations of semi-simple Lie groups, Ann. of Math. (2), 52, 509-517 (1950) · Zbl 0045.30901 |
| [35] | Skudlarek H.-L., Die unzerlegbaren Charaktere einiger diskreter Gruppen, Math. Ann. 223 (1976), 213-231.; Skudlarek, H.-L., Die unzerlegbaren Charaktere einiger diskreter Gruppen, Math. Ann., 223, 213-231 (1976) · Zbl 0313.43010 |
| [36] | Stuck G. and Zimmer R., Stabilizers for ergodic actions of higher rank semisimple groups, Ann. of Math. (2) 139 (1994), no. 3, 723-747.; Stuck, G.; Zimmer, R., Stabilizers for ergodic actions of higher rank semisimple groups, Ann. of Math. (2), 139, 3, 723-747 (1994) · Zbl 0836.22018 |
| [37] | Takesaki M., Theory of operator algebras. I, Encyclopaedia Math. Sci. 124, Springer-Verlag, Berlin 2002.; Takesaki, M., Theory of operator algebras. I (2002) · Zbl 0990.46034 |
| [38] | Thoma E., Die unzerlegbaren, positiv-definiten Klassenfunktionen der abzählbar unendlichen, symmetrischen Gruppe, Math. Z. 85 (1964), 40-61.; Thoma, E., Die unzerlegbaren, positiv-definiten Klassenfunktionen der abzählbar unendlichen, symmetrischen Gruppe, Math. Z., 85, 40-61 (1964) · Zbl 0192.12402 |
| [39] | Thoma E., Über unitäre Darstellungen abzählbarer, diskreter Gruppen, Math. Ann. 153 (1964), 111-138.; Thoma, E., Über unitäre Darstellungen abzählbarer, diskreter Gruppen, Math. Ann., 153, 111-138 (1964) · Zbl 0136.11603 |
| [40] | Thoma E., Invariante positiv definite Klassenfunktionen und ergodische Maße, Math. Ann. 162 (1965/1966), 172-189.; Thoma, E., Invariante positiv definite Klassenfunktionen und ergodische Maße, Math. Ann., 162, 172-189 (19651966) · Zbl 0151.18801 |
| [41] | Umegaki H., Conditional expectation in an operator algebra, Tôhoku Math. J. (2) 6 (1954), 177-181.; Umegaki, H., Conditional expectation in an operator algebra, Tôhoku Math. J. (2), 6, 177-181 (1954) · Zbl 0058.10503 |
| [42] | Vershik A. M., Nonfree actions of countable groups and their characters, J. Math. Sci. (N. Y.) 174 (2011), no. 1, 1-6.; Vershik, A. M., Nonfree actions of countable groups and their characters, J. Math. Sci. (N. Y.), 174, 1, 1-6 (2011) · Zbl 1279.37004 |
| [43] | Vershik A. M. and Kerov S. V., Characters and factor representations of the infinite symmetric group, Sov. Math. Dokl. 23 (1981), 389-392.; Vershik, A. M.; Kerov, S. V., Characters and factor representations of the infinite symmetric group, Sov. Math. Dokl., 23, 389-392 (1981) · Zbl 0534.20008 |
| [44] | Voiculescu D., Représentations factorielles de type \({\text{II}_1}\) de \({U(\infty)} \), J. Math. Pures Appl. (9) 55 (1976), 1-20.; Voiculescu, D., Représentations factorielles de type \({\text{II}_1}\) de \({U(\infty)} \), J. Math. Pures Appl. (9), 55, 1-20 (1976) · Zbl 0352.22014 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
