Abstract
Let G denote a semisimple group, Γ a discrete subgroup, B=G/P the Poisson boundary. Regarding invariants of discrete subgroups we prove, in particular, the following:
(1) For any Γ-quasi-invariant measure η on B, and any probablity measure μ on Γ, the norm of the operator πη(μ) on L 2(B,η) is equal to ∥λΓ(μ)∥, where πη is the unitary representation in L 2(X,η), and λΓ is the regular representation of Γ.
(2) In particular this estimate holds when η is Lebesgue measure on B, a Patterson–Sullivan measure, or a μ-stationary measure, and implies explicit lower bounds for the displacement and Margulis number of Γ (w.r.t. a finite generating set), the dimension of the conformal density, the μ-entropy of the measure, and Lyapunov exponents of Γ.
(3) In particular, when G=PSL2(ℂ) and Γ is free, the new lower bound of the displacement is somewhat smaller than the Culler–Shalen bound (which requires an additional assumption) and is greater than the standard ball-packing bound.
We also prove that ∥πη(μ)∥=∥λG(μ)∥ for any amenable action of G and μ∈L 1(G), and conversely, give a spectral criterion for amenability of an action of G under certain natural dynamical conditions. In addition, we establish a uniform lower bound for the μ-entropy of any measure quasi-invariant under the action of a group with property T, and use this fact to construct an interesting class of actions of such groups, related to 'virtual' maximal parabolic subgroups. Most of the results hold in fact in greater generality, and apply for instance when G is any semi-simple algebraic group, or when Γ is any word-hyperbolic group, acting on their Poisson boundary, for example.
Similar content being viewed by others
References
Adams, S.: Boundary amenability for hyperbolic groups and an application to smooth dynamics of simple groups, Topology 33 (1994), 765–783.
Albuquerque, P.: Patterson-Sullivan theory in higher rank symmetric spaces, Geom. Funct. Anal. 9 (1999), 1–28.
Anantharaman-Delaroche, C.: On spectral characterizations of amenability, Preprint 2001.
Anantharaman-Delaroche, C. and Renault, J.: Amenable Groupoids, Enseign. Math. Monogr. Ser. 36, 2000.
Billingsley, P.: Probability and Measure, 2nd edn, Wiley, New York, 1986.
Besson, G., Courtois, G. and Gallot, S.: Volume et entropie minimale des espaces localement symmétrique, Invent. Math. 103 (1991), 417–445.
Cowling, M.: Sur les coefficients des représentations unitaires des groupes de Lie simples, In: Lecture Notes in Math. 739, Springer, New York, 1979, pp. 132–178.
Connes, A., Feldman, J. and Weiss, B.: An amenable equivalence relation is generated by a single transformation. Ergodic Theory Dynam. Systems 1 (1981), 431–450.
Cowling, M., Haagerup, U. and Howe, R.: Almost L 2 matrix coefficients, J. Reine Angew. Math. 387 (1988), 97–110.
Culler, M. and Shalen, P. B.: Paradoxical decompositions, 2-generator Kleinian groups, and volumes of hyperbolic 3-manifolds, J. Amer. Math. Soc. 5(2) (1992), 231–289.
Furman, A.: Random walks on groups and random transformations, In: A. Katok and B. Hasselblatt (eds), Handbook of Dynamical Systems, vol. I. To appear.
Furstenberg, H.: A Poisson formula for semi-simple Lie groups, Ann. of Math. (2) 77 (1963), 335–386.
Furstenberg, H.: Noncommuting random products, Trans. Amer. Math. Soc. 108 (1963), 377–428.
Guivarc'h, Y.: Sur loi des grands nombres et le rayon spectral d'une marche aléatoires, Asterisque 74 (1980), 47–98.
Guivarc'h, Y.: Produits de matrices aléatoires et applications aux propriétés géomtriques des sous-groupes du groupe linéare, Ergodic Theory Dynam. Systems 10 (1990), 483–512.
Guivarc'h, Y. and Raugi, A.: Products of random matrices: convergence theorems, In: Contemporary Math. 50, Amer. Math. Soc., Providence, 1986, pp. 31–53.
Guivarc'h, Y., Ji, L. and Taylor, J. C.: Compactifications of Symmetric Spaces, Progr. in Math. 156, Birkhäuser, Basel, 1998.
Gangolli, R. and Varadarajan, V. S.: Harmonic Analysis of Spherical Functions on Real Reductive Groups, Modern Surv. Math. 101, Springer, New York, 1988.
de la Harpe, P. and Valette, A.: Propriété (T) de Kazhdan pour les groupes localement compacts, Astérisque 175, Soc. Math. de France, 1989.
Helgason, S.: Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, New York, 1978.
Helgason, S.: Groups and Geometric Analysis, Academic Press, New York, 1984.
Herz, C. S.: Sur le phénomène de Kunze-Stein. C.R. Acad. Sci. Paris 271 (1970), 491–493.
Hou, Y.: Geometrically infinite negatively curved three manifolds, PhD Thesis, University of Illinois at Chicago, 2000.
Howe, R. and Tan, E. C.: Non-Abelian Harmonic Analysis, Springer, New York, 1992.
Kaimanovich, V. A. and Vershik, A.: Random walks on discrete groups: Boundary and entropy, Ann. Probab. 11 (1983), 457–490.
Kaimanovich, V. A.: The Poisson boundary of groups with hyperbolic properties, Ann. of Math. 152 (2000), 659–692.
Kaimanovich, V. A.: The Poisson boundary of covering Markov operators, Israel J. Math. 89 (1995), 77–134.
Kaimanovich, V. A.: Invariant measures of the geodesic flow and measures at infinity on negatively curved manifolds, Ann. Inst. H. Poincaré 53 (1990), 361–393.
Kuhn, M. G.: Amenable actions and weak containment of certain representations of discrete groups. Proc. Amer. Math. Soc. 122(3) (1994), 751–757.
Kullback, S.: Information Theory and Statistics, 2nd edn, Dover, New York, 1968.
Ledrappier, F.: Poisson boundaries of discrete groups of matrices, Israel J. Math. 50 (1985), 319–336.
Ledrappier, F.: Quelques propriétes des exposants charactéristiques, In: Lecture Notes in Math. 1097, Springer, New York, 1982, pp. 306–396.
Ledrappier, F.: Structure au bord des variétés à courbure négative, Séminare de théorie spectrale et géométrie, Grenoble, 1994, pp. 97–122.
Ledrappier, F.: Harmonic measures and Bowen-Margulis measures, Israel J. Math. 71 (1990), 275–287.
Ledrappier, F.: A heat kernel characterization of asymptotic harmonicity, Proc. Amer. Math. Soc. 118 (1993), 1001–1004.
Ledrappier, F.: Sharp estimates for the entropy, In: M. Picardello (ed.), Proc. Internat. Meeting Frascati, 1–10 July 1991, Plenum Press, New York, 1992, pp. 281–288.
Margulis, G. A.: Discrete Subgroups of Semisimple Lie Groups, Modern Surv. Math, 17, Springer, New York, 1991.
Margulis, G. A.: Discrete subgroups of motions of manifolds of non-positive curvature, Amer. Math. Soc. Trans. 109 (1977), 33–45.
Nevo, A.: Amenable actions and actions with property T, MSc Thesis, Hebrew University, 1987 (in Hebrew).
Nevo, A.: A note on property T and factors of Poisson boundaries, Landau Foundation Preprint Series, No. 19, Hebrew University, 1991.
Nevo, A.: Spectral transfer and pointwise ergodic theorems for semi-simple Kazhdan groups, Math. Res. Lett. 5 (1998), 1–21.
Nevo, A.: Group actions with positive μ-entropy, Preprint.
Nevo, A.: Boundary theory and harmonic analysis on boundary-transitive graphs, Amer. J. Math. 116 (1994), 243–282.
Nevo, A.: Displacement of discrete subgroups of semisimple groups, In preparation.
Nevo, A. and Zimmer, R. J.: Homogeneous projective factors for actions of semi-simple Lie groups, Invent. Math. 138 (1999), 229–252.
Nevo, A. and Zimmer, R. J.:Rigidity of Furstenberg entropy for semi-simple Lie group actions. Ann. Sci. École. Norm. Sup. 33 (2000), 321–343.
Nevo, A. and Zimmer, R. J.: A structure theorem for actions of semisimple Lie groups. Ann. of Math. 157 (2002), 1–30.
Oh, H.: Uniform pointwise bounds for matrix coefficients of unitary representations and applications to Kazhdan's constants, Duke Math. J. 113 (2002), 133–192.
Patterson, S. J.: The limit set of a Fuchsian group, Acta Math. 136 (1976), 241–273.
Spatzier, R. and Zimmer, R. J.: Fundamental groups of negatively curved manifolds and actions of semisimple groups, Topology 30(4) (1991), 591–601.
Sullivan, D.: Discrete conformal groups and measurable dynamics, Bull. Amer. Math. Soc. 6 (1982), 57–63.
Sullivan, D.: The density at infinity of a discrete group of hyperbolic motions, Publ. Math. IHES 50 (1979), 171–202.
Virtser, D. A.: Products of random matrices and operators, Theory Probab. Appl. 24 (1979), 367–377.
Zimmer, R. J.: Ergodic Theory and Semi-Simple Groups, Birkhäuser, Boston, 1984.
Zimmer, R. J.: Amenable ergodic group actions and an application to Poisson boundaries of Random walks. J. Funct. Anal. 27 (1978), 350–372.
Zimmer, R. J.: Induced and amenable actions of Lie groups. Ann. Sci. École Norm. Sup. 11 (1978), 407–428.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Nevo, A. The Spectral Theory of Amenable Actions and Invariants of Discrete Groups. Geometriae Dedicata 100, 187–218 (2003). https://doi.org/10.1023/A:1025839828396
Issue Date:
DOI: https://doi.org/10.1023/A:1025839828396