Amenable ergodic group actions and an application to Poisson boundaries of random walks

Zimmer, Robert J.

doi:10.1016/0022-1236(78)90013-7

Cited by 194 publications

(165 citation statements)

References 11 publications

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“…For the general background of these definitions we refer to [5], [13] and [16]. For A = C and for a measure preserving action of G on (X, y, /i)we define further cohomology groups…”

Section: Z\g U{xfia))/b\g U(xfia))mentioning

confidence: 99%

Amenability, Kazhdan's property T, strong ergodicity and invariant means for ergodic group-actions

Schmidt

1981

Ergod. Th. Dynam. Sys.

122

110

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Abstract. This paper discusses the relations between the following properties of finite measure preserving ergodic actions of a countable group G: strong ergodicity (i.e. the non-existence of almost invariant sets), uniqueness of G-invariant means on the measure space carrying the group action, and certain cohomological properties. Using these properties one can characterize all actions of amenable groups and of groups with Kazhdan's property T. For groups which fall in between these two definitions these notions lead to some interesting examples.

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“…For the general background of these definitions we refer to [5], [13] and [16]. For A = C and for a measure preserving action of G on (X, y, /i)we define further cohomology groups…”

Section: Z\g U{xfia))/b\g U(xfia))mentioning

confidence: 99%

Amenability, Kazhdan's property T, strong ergodicity and invariant means for ergodic group-actions

Schmidt

1981

Ergod. Th. Dynam. Sys.

122

110

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“…Moreover, it will be essential for our purposes that the action of Λ on the Poisson boundary B is amenable with respect to the measure ν, [69].…”

Section: Tight Homomorphismsmentioning

confidence: 99%

Maximal Representations of Surface Groups: Symplectic Anosov Structures

Burger¹,

Iozzi²,

Labourie³

et al. 2005

Pure and Applied Mathematics Quarterly

167

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Abstract. Let G be a connected semisimple Lie group such that the associated symmetric space X is Hermitian and let Γ g be the fundamental group of a compact orientable surface of genus g ≥ 2.We survey the study of maximal representations of Γ g into G, that is the subset of Hom(Γ g , G) characterized by the maximality of the Toledo invariant ([17] and [15]). Then we concentrate on the particular case G = Sp(2n, R), and we show that if ρ is any maximal representation then the image ρ(Γ g ) is a discrete, faithful realizations of Γ g as a Kleinian group of complex motions in X with an associated Anosov system, and whose limit set in an appropriate compactification of X is a rectifiable circle.

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“…In 1978 R. Zimmer [8] introduced the notion of amenable action for locally compact topological groups G. When X is a homogeneous G space, say X = G/H, amenability of the action is equivalent to the amenability of H.…”

Section: Introductionmentioning

confidence: 99%

Transference for amenable actions

Hebisch¹,

Kuhn²

2004

Proc. Amer. Math. Soc.

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Abstract. Suppose G acts amenably on a measure space X with quasiinvariant σ-finite measure m. Let σ be an isometric representation of G on L p (X, dm) and µ a finite Radon measure on G. We show that the operator σµf (x) = G (σ(g)f )(x)dµ(g) has L p (X, dm)-operator norm not exceeding the L p (G)-operator norm of the convolution operator defined by µ. We shall also prove an analogous result for the maximal function M associated to a countable family of Radon measures µn.

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Amenable ergodic group actions and an application to Poisson boundaries of random walks

Cited by 194 publications

References 11 publications

Amenability, Kazhdan's property T, strong ergodicity and invariant means for ergodic group-actions

Amenability, Kazhdan's property T, strong ergodicity and invariant means for ergodic group-actions

Maximal Representations of Surface Groups: Symplectic Anosov Structures

Transference for amenable actions

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