Invariant random subgroups of lamplighter groups

详细信息    查看全文

• 作者：Lewis Bowen ; Rostislav Grigorchuk ; Rostyslav Kravchenko
• 刊名：Israel Journal of Mathematics
• 出版年：2015
• 出版时间：April 2015
• 年：2015
• 卷：207
• 期：2
• 页码：763-782
• 全文大小：259 KB
• 参考文献：[AB+11]M. Abert, N. Bergeron, I. Biringer, T. Gelander, N. Nikolov, J. Raimbault and I. Samet, On the growth of Betti numbers of locally symmetric spaces, Comptes Rendus Mathématique. Académie des sciences. Paris 349 (2011), 831-35.MATH MathSciNet View Article
  [AB+12]M. Abert, N. Bergeron, I. Biringer, T. Gelander, N. Nikolov, J. Raimbault and I. Samet, On the growth of L 2-invariants for sequences of lattices in Lie groups, arXiv:1210.2961
  [AGV12]M. Abert, Y. Glasner and B. Virag, Kesten’s theorem for invariant random subgroups, Duke Mathematical Journal 163 (2014), 465-88.MATH MathSciNet View Article
  [Bo10]L. Bowen, Random walks on coset spaces with applications to Furstenberg entropy, Inventiones Mathematicae 196 (2014), 485-10.MATH MathSciNet View Article
  [Bo12]L. Bowen, Invariant random subgroups of the free group, Groups, Geometry and Dynamics, to appear.
  [BS06]U. Bader and Y. Shalom, Factor and normal subgroup theorems for lattices in products of groups, Inventiones Mathematicae 163 (2006), 415-54MATH MathSciNet View Article
  [BV93]M. E. B. Bekka and A. Valette, Kazhdan’s property (T) and amenable representations, Mathematische Zeitschrift 212 (1993), 293-99.MATH MathSciNet View Article
  [DS02]S. G. Dani, On conjugacy classes of closed subgroups and stabilizers of Borel actions of Lie groups, Ergodic Theory and Dynamical Systems 22 (2002), 1697-714.MATH MathSciNet View Article
  [DM11]A. Dudko and K. Medynets, On characters of inductive limits of symmetric groups, Journal of Functional Analysis 264 (2013), 1565-598.MATH MathSciNet View Article
  [DM12]A. Dudko and K. Medynets, Finite factor representations of Higman-Thompson groups, Groups, Geometry, and Dynamics 8 (2014), 375-89.MATH MathSciNet View Article
  [GK12]R. Grigorchuk and R. Kravchenko, On the lattice of subgroups of the lamplighter group, International Journal of Algebra and Computation 24 (2014), to appear.
  [Gr11]R. Grigorchuk, Some topics of dynamics of group actions on rooted trees, Proceedings of the Steklov Institute of Mathematics 273 (2011), 64-75.MATH MathSciNet View Article
  [GS99]V. Golodets and S. D. Sinelshchikov, On the conjugacy and isomorphism problems for stabilizers of Lie group actions, Ergodic Theory and Dynamical Systems 19 (1999), 391-11.MATH MathSciNet View Article
  [GW97]E. Glasner and B. Weiss, Kazhdan’s property T and the geometry of the collection of invariant measures, Geometric and Functional Analysis 7 (1997), 917-35.MATH MathSciNet View Article
  [JS87]V. F. R. Jones and K. Schmidt, Asymptotically Invariant Sequences and Approximate Finiteness, American Journal of Mathematics 109 (1987), 91-14.MATH MathSciNet View Article
  [Kro85]P. H. Kropholler, A note on the cohomology of metabelian groups, Mathematical Proceedings of the Cambridge Philosophical Society 98 (1985), 437-45.MATH MathSciNet View Article
  [LOS78]J. Lindenstrauss, G. H. Olsen and Y. Sternfeld, The Poulsen simplex, Université de Grenoble. Annales de l’Institut Fourier 28 (1978), 91-14.MATH MathSciNet View Article
  [Ol79]G. H. Olsen, On simplices and the Poulsen simplex, in Functional Analysis: Surveys and Recent Results, II (Proc. Second Conf. Functional Anal., Univ. Paderborn, Paderborn, 1979), Notas de Matemática, Vol. 68, North-Holland, Amsterdam-New York, 1980, pp. 31-2.
  [Sa11]D. Savchuk, Schreier graphs of actions of Thompson’s group F on the unit interval and on the Cantor set, Geometriae Dedicata (2014), to appear.
  [Sch84]K. Schmidt, Asymptotic properties of unitary representations and mixing, Proceedings of the London Mathematical Society 48 (1984), 445-60.MATH MathSciNet View Article
  [SZ94]G. Stuck and R. J. Zimmer, Stabilizers for ergodic actions of higher rank semisimple groups, Annals of Mathematics 139 (1994), 723-47.MATH MathSciNet View Article
  [Ve10]A. Vershik, Nonfree actions of countable groups and their characters, Zapiski Nauchnykh Seminarov (POMI) 378 (2010), 5-6; English translation: Journal of Mathematical Sciences 174 (2011), 1-.
  [Ve11]A. Vershik, Totally nonfree actions and infinite symmetric group, Moscow Mathematical Journal 12 (2012), 193-12.MATH MathSciNet
  [Vo12]Y. Vorobets, Notes on the Schreier graphs of the Grigorchuk group, Contemporary Mathematics 567 (2012), 221-48.MathSciNet
  
• 作者单位：Lewis Bowen (1)
  Rostislav Grigorchuk (2)
  Rostyslav Kravchenko (1)
  
  1. Mathematics Department, University of Texas at Austin, 1 University Station C1200, Texas, 78712, USA
  2. Department of Mathematics, Mailstop 3368, Texas A&M University, College Station, TX, 77843, USA
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Algebra
  Group Theory and Generalizations
  Analysis
  Applications of Mathematics
  Mathematical and Computational Physics
  
• 出版者：Hebrew University Magnes Press
• ISSN：1565-8511

文摘

Let G be one of the lamplighter groups \({({\Bbb Z}/p{\Bbb Z})^n} \wr {\Bbb Z}\) and Sub(G) the space of all subgroups of G. We determine the perfect kernel and Cantor-Bendixson rank of Sub(G). The space of all conjugation-invariant Borel probability measures on Sub(G) is a simplex. We show that this simplex has a canonical Poulsen subsimplex whose complement has only a countable number of extreme points. If F is a finite group and Γ an infinite group which does not have property (T), then the conjugation-invariant probability measures on Sub(\(F \wr \Gamma \)) supported on \(_{ \oplus \Gamma }F\) also form a Poulsen simplex.