Kazhdan and Haagerup properties from the median viewpoint

Abstract We prove the existence of a close connection between spaces with measured walls and median metric spaces. We then relate properties (T) and Haagerup (a-T-menability) to actions on median spaces and on spaces with measured walls. This allows us to explore the relationship between the classical properties (T) and Haagerup and their versions using affine isometric actions on L p -spaces. It also allows us to answer an open problem on a dynamical characterization of property (T), generalizing results of Robertson and Steger.

[1]  J. Wells,et al.  Embeddings and Extensions in Analysis , 1975 .

[2]  P. Pansu.  Cohomologie L : invariance sous quasiisométries , 2004 .

[3]  M. Bożejko.  Positive-definite kernels, length functions on groups and a noncommutative von Neumann inequality , 1989 .

[4]  Roger C. Alperin.  Locally compact groups acting on trees and propertyT , 1982 .

[5]  Peter Scott,et al.  Minimal Cubings , 2005, Int. J. Algebra Comput..

[6]  W. Ballmann,et al.  On L2-cohomology and Property (T) for Automorphism Groups of Polyhedral Cell Complexes , 1997 .

[7]  Graham Niblo,et al.  From Wall Spaces to Cat(0) Cube Complexes , 2005, Int. J. Algebra Comput..

[8]  V. Guirardel.  Actions of finitely generated groups on $\mathbb{R}$-trees , 2008 .

[9]  D. Jacob Wildstrom.  Cost thresholds for dynamic resource location , 2008, Discret. Appl. Math..

[10]  M. van de Vel.  Dimension of Binary Convex Structures , 1984 .

[11]  P. Delorme.  1-cohomologie des représentations unitaires des groupes de Lie semi-simples et résolubles. Produits tensoriels continus de représentations , 1977 .

[12]  Marshall C. Yovits,et al.  Ohio State University , 1974, SGAR.

[13]  S. Popa,et al.  On a class of $\mathrm{II}_1$ factors with at most one Cartan subalgebra , 2007, 0706.3623.

[14]  W. B. Johnson,et al.  l~p (p > 2) does not coarsely embed into a Hilbert space , 2004 .

[15]  B. Nica.  Group actions on median spaces , 2008, 0809.4099.

[16]  Fred S. Roberts,et al.  The Median Procedure on Median Graphs , 1998, Discret. Appl. Math..

[17]  THE DUAL OF THE CATEGORY OF GENERALIZED TREES , 2004 .

[18]  Group actions on Banach spaces and a geometric characterization of a-T-menability , 2004, math/0404402.

[19]  M. Sholander.  Medians, lattices, and trees , 1954 .

[20]  Graham A. Niblo,et al.  Groups acting on cubes and Kazhdan’s property (T) , 1998 .

[21]  Frederic Paulin,et al.  Simplicite de groupes d'automorphismes d'espaces a courbure negative , 1998, math/9812167.

[22]  N. Monod,et al.  Property (T) and rigidity for actions on Banach spaces , 2005, math/0506361.

[23]  Z. Sela.  Acylindrical accessibility for groups , 1997 .

[24]  Srishti D. Chatterji,et al.  Positive Definite Kernels , 1983 .

[25]  G. Birkhoff,et al.  A ternary operation in distributive lattices , 1947 .

[26]  V. Guirardel,et al.  Actions of finitely generated groups on R-trees , 2006, math/0607295.

[27]  Weak amenability of CAT(0)-cubical groups , 2007, math/0702568.

[28]  Vincent Lafforgue,et al.  Un renforcement de la propriété (T) , 2008 .

[29]  Negative definite kernels and a dynamical characterization of property (T) for countable groups , 1998, Ergodic Theory and Dynamical Systems.

[30]  J. Krivine,et al.  Lois stables et espaces $L^p$ , 1967 .

[31]  A. Guichardet.  Etude de la 1-cohomologie et de la topologie du dual pour les groupes de Lie à radical Abelien , 1977 .

[32]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[33]  D. Garling.  AN F‐SPACE SAMPLER (London Mathematical Society Lecture Note Series, 89) , 1985 .

[34]  Cohomologie ℓp espaces de Besov , 2003 .

[35]  A. Valette,et al.  Isometric Group Actions on Banach Spaces and Representations Vanishing at Infinity , 2006, math/0612398.

[36]  Van de M. L. J. Vel.  Theory of convex structures , 1993 .

[37]  Juhani Nieminen.  The ideal structure of simple ternary algebras , 1978 .

[38]  A. Valette,et al.  Proper actions of lamplighter groups associated with free groups , 2007, 0707.2039.

[39]  L. Nachbin,et al.  The Haar integral , 1965 .

[40]  Michel Deza,et al.  Metric subspaces of L[1] , 1982 .

[41]  Nigel J. Kalton,et al.  An F-space sampler , 1984 .

[42]  Eric R. Verheul.  Multimedians in metric and normed spaces , 1993 .

[43]  Michel Deza,et al.  Hypermetrics in geometry of numbers , 1993, Combinatorial Optimization.

[44]  Hyperbolic groups admit proper affine isometric actions on lp-spaces , 2004, math/0411234.

[45]  Michah Sageev,et al.  Ends of Group Pairs and Non‐Positively Curved Cube Complexes , 1995 .

[46]  J. Faraut,et al.  Distances hilbertiennes invariantes sur un espace homogene , 1974 .

[47]  A. Valette,et al.  Spaces with measured walls, the Haagerup property and property (T) , 2004, Ergodic Theory and Dynamical Systems.

[48]  Hans-Jürgen Bandelt,et al.  Median algebras , 1983, Discret. Math..

[49]  I. J. Schoenberg,et al.  Metric spaces and positive definite functions , 1938 .

[50]  H. Bandelt,et al.  Metric graph theory and geometry: a survey , 2006 .

[51]  Mladen Bestvina,et al.  Stable actions of groups on real trees , 1995 .

[52]  A. Valette.  Crofton formulae and geodesic distance in hyperbolic spaces , .

[53]  Victor Chepoi,et al.  Graphs of Some CAT(0) Complexes , 2000, Adv. Appl. Math..

[54]  M. Sholander.  MEDIANS AND BETWEENNESS , 1954 .

[55]  Patrice Assouad.  Sur les Inégalités Valides dans L1 , 1984, Eur. J. Comb..

[56]  Bogdan Nica.  Cubulating spaces with walls , 2004 .

[57]  C. Akemann,et al.  Unbounded Negative Definite Functions , 1981, Canadian Journal of Mathematics.

[58]  A. Valette,et al.  Groups with the Haagerup Property: Gromov’s a-T-menability , 2001 .

[59]  Alain Valette,et al.  Kazhdan's Property (T) , 2008 .

[60]  Y. Watatani.  Property T of Kazhdan implies property FA of Serre , 1982 .

[61]  Groups acting on CAT(0) cube complexes , 1997, math/9702231.

[62]  Rudolf Scharlau.  A Characterization of Tits Buildings by Metrical Properties , 1985 .

[63]  Wilfried Imrich,et al.  Recognizing Median Graphs in Subquadratic Time , 1999, Theor. Comput. Sci..

[64]  H. M. Mulder.  The interval function of a graph , 1980 .

[65]  C. Houghton.  Ends of locally compact groups and their coset spaces , 1974, Journal of the Australian Mathematical Society.

[66]  Pierre de la Harpe,et al.  La propriété (T) de Kazhdan pour les groupes localement compacts , 1989 .