The first <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msup><mml:mi>L</mml:mi><mml:mi>p</mml:mi></mml:msup></mml:math>-cohomology of some finitely generated groups and p-harmonic functions

Puls, Michael J.

doi:10.1016/j.jfa.2006.04.011

Cited by 14 publications

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“…In the context of some classes of metric spaces, L p -cohomology is a quasi-isometry invariant with interesting applications to classification problems. This notion is defined, for example, for simplicial complexes [7,14,25,21], Riemannian manifolds [18,19,34,31], discrete and topological groups [3,4,5,6,12,30,35,36] and more general metric measure spaces [16,33,36].…”

Section: Introductionmentioning

confidence: 99%

Relative L^p-cohomology and application to Heintze groups

Sequeira

2024

Ann. Fenn. Math.

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We introduce the notion of relative \(L^p\)-cohomology as a quasi-isometry invariant defined for a Gromov-hyperbolic space and a point on its boundary at infinity and reproduce some basic properties of \(L^p\)-cohomology in this context. In the case of degree 1 we show a relation between the relative and the classical \(L^p\)-cohomology. As an application, we explicitly construct non-zero relative \(L^p\)-cohomology classes for a purely real Heintze group of the form \(\mathbb{R}^{n-1}\rtimes_\alpha\mathbb{R}\), which gives a way to prove that the eigenvalues of \(\alpha\), up to a scalar multiple, are invariant under quasi-isometries.

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Section: Introductionmentioning

confidence: 99%

Relative L^p-cohomology and application to Heintze groups

Sequeira

2024

Ann. Fenn. Math.

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“…This of course would resolve Gromov's conjecture. More information about the first reduced L p -cohomology (and the special case of L 2 -cohomology) can be found in [Pansu 1989;2008;Tessera 2009] for various manifolds, and in [Bekka and Valette 1997;Bourdon 2004;Bourdon et al 2005;Elek 1998;Martin and Valette 2007;Puls 2003;2006; for finitely generated groups. As implied earlier, there is a strong connection between the vanishing of the first reduced L p -cohomology and the nonexistence nonconstant p-harmonic functions; for a proof in the case of homogeneous Riemannian manifolds, see [Tessera 2009, Proposition 4.11].…”

Section: Introductionmentioning

confidence: 99%

Graphs of bounded degree and thep-harmonic boundary

Puls¹

2010

Pacific J. Math.

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Let p be a real number greater than one and let G be a connected graph of bounded degree. We introduce the p-harmonic boundary of G and use it to characterize the graphs G for which the constant functions are the only pharmonic functions on G. We show that any continuous function on the pharmonic boundary of G can be extended to a function that is p-harmonic on G. We also give some properties of this boundary that are preserved under rough-isometries. Now let be a finitely generated group. As an application of our results, we characterize the vanishing of the first reduced p -cohomology of in terms of the cardinality of its p-harmonic boundary. We also study the relationship between translation invariant linear functionals on a certain difference space of functions on , the p-harmonic boundary of , and the first reduced p -cohomology of .

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Linear and Nonlinear Harmonic Boundaries of Graphs; An Approach with ℓ p-Cohomology in Degree One

Gournay

2020

Frontiers in Analysis and Probability

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The first Lp-cohomology of some finitely generated groups and p-harmonic functions

Cited by 14 publications

References 12 publications

Relative L^p-cohomology and application to Heintze groups

Relative L^p-cohomology and application to Heintze groups

Graphs of bounded degree and thep-harmonic boundary

Linear and Nonlinear Harmonic Boundaries of Graphs; An Approach with ℓ p-Cohomology in Degree One

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