Ergodic boundary representations

Boyer, Adrien; Link, Guido; Pittet, Ch.

doi:10.1017/etds.2017.118

Cited by 6 publications

(8 citation statements)

References 31 publications

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“…Comments on the uniform boundedness condition. The following theorem, which generalizes ideas from [BLP17], gives sufficient conditions for the uniform boundedness condition to hold.…”

Section: Statement Of the Resultsmentioning

confidence: 99%

“…The uniform boundedness condition has been studied in [BM11], [Boy16], [Gar14], [BLP17] and [BPL17] where authors investigate generalizations of the von Neumann ergodic theorem to the situation where the measure is only quasi-invariant. In several cases of interest, the relevant generalization of von Neumann means are the normalized means 1…”

Section: Statement Of the Resultsmentioning

confidence: 99%

“…space X with non-arithmetic spectrum and a finite BMS measure, and L is the length function associated to the action and B is the Gromov boundary and ν is the Patterson-Sullivan probability measure; • [BLP17] if G is a noncompact, connected, semisimple Lie group with finite center, let us keep the notation from section 2.1.2: Λ is a lattice in G, B := G/P , ν is a quasi-invariant probability measure and L is defined by the formula…”

Section: Statement Of the Resultsmentioning

confidence: 99%

See 2 more Smart Citations

Radial rapid decay does not imply rapid decay

Boyer¹,

Lobos²,

Pittet³

2020

Preprint

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“…Comments on the uniform boundedness condition. The following theorem, which generalizes ideas from [BLP17], gives sufficient conditions for the uniform boundedness condition to hold.…”

Section: Statement Of the Resultsmentioning

confidence: 99%

Section: Statement Of the Resultsmentioning

confidence: 99%

Section: Statement Of the Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Radial rapid decay does not imply rapid decay

Boyer¹,

Lobos²,

Pittet³

2020

Preprint

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show abstract

“…The following theorem, which generalizes ideas from [6], gives sufficient conditions for the uniform boundedness condition to hold. Theorem 2.8 (Sufficient conditions for the uniform boundedness condition).…”

Section: Statement Of the Criterionmentioning

confidence: 99%

Radial rapid decay does not imply rapid decay

Boyer¹,

Lobos²,

Pittet³

2023

Annales De l'Institut Fourier

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We provide a new, dynamical criterion for the radial rapid decay property. We work out in detail the special case of the group Γ := SL 2 (A), where A := Fq[X, X −1 ] is the ring of Laurent polynomials with coefficients in Fq, endowed with the length function coming from a natural action of Γ on a product of two trees, and show that it has the radial rapid decay (RRD) property and doesn't have the rapid decay (RD) property. We show that the criterion also applies to all irreducible lattices (uniform or not) in semisimple Lie groups with finite center endowed with a length function defined with the help of a Finsler metric. When the rank is greater or equal to two and the lattice is non-uniform, the lattice has RRD but not RD. These examples answer a question asked by Chatterji and moreover show that, unlike the RD property, the RRD property isn't inherited by open subgroups.Résumé. -Nous établissons un nouveau critère dynamique entraînant la propriété de décroissance rapide radiale. Nous explicitons le cas particulier du groupe Γ := SL 2 (A), où A := Fq[X, X −1 ] est l'anneau des polynômes de Laurent à coefficients dans le corps fini Fq, muni d'une fonction longueur provenant d'une action naturelle de Γ sur le produit de deux arbres. Nous prouvons que pour cette fonction longueur, ce groupe vérifie la propriété de décroissance rapide radiale (RRD), mais ne vérifie pas la propriété de décroissance rapide (RD). Nous prouvons aussi que notre critère s'applique à tout réseau irréductible (uniforme ou non), de tout groupe de Lie semi-simple à centre fini, muni d'une certaine fonction longueur définie à l'aide d'une métrique de Finsler. Lorsque le rang réel est supérieur ou égal à deux et que le réseau n'est pas uniforme, le réseau vérifie la propriété RRD, mais pas la propriété RD. Ces exemples répondent à une question de Chatterji et montrent que, contrairement à la propriété RD, la propriété RRD n'est pas héréditaire par passage à un sous-groupe ouvert.

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“…The representation π 0 is unitary and might be thought as an analog of the endpoint of principal series for SL(2, R). It appears in different contexts as a boundary representation and it is also called quasi-regular representation (or Koopman representation)see [18], [3], [32], [10], [12], [13], [15], [14], [29], [41] and [42] for boundary representations and see [24] and [25] for other quasi-regular representations. More recently in [19], boundary representations of hyperbolic groups have been used to obtain results in the general setting of locally compact groups.…”

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confidence: 99%

Some spherical functions on hyperbolic groups

Boyer

2020

J. Topol. Anal.

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We investigate properties of some spherical fonctions defined on hyperbolic groups using boundary representations on the Gromov boundary endowed with the Patterson-Sullivan measure class. We prove sharp decay estimates for spherical functions as well as spectral inequalities associated with boundary representations. This point of view on the boundary allows us to view the so-called property RD (also called Haagerup's inequality) as a particular case of a more general behavior of spherical functions on hyperbolic groups. In particular, we give a constructive proof using a boundary unitary representation of a result due to de la Harpe and Jolissaint asserting that hyperbolic groups satisfy property RD. Finally, we prove that the family of boundary representations studied in this paper, which can be regarded as a one parameter deformation of the boundary unitary representation, are slow growth representations acting on a Hilbert space admitting a proper 1-cocycle.

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Ergodic boundary representations

Abstract: We prove a von Neumann type ergodic theorem for averages of unitary operators arising from the Furstenberg-Poisson boundary representation (the quasi-regular representation) of any lattice in a non-compact connected semisimple Lie group with finite center.

Cited by 6 publications

References 31 publications

Radial rapid decay does not imply rapid decay

Radial rapid decay does not imply rapid decay

Radial rapid decay does not imply rapid decay

Some spherical functions on hyperbolic groups

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