Conditionally negative type functions on groups acting on regular trees

Gournay, Antoine; Jolissaint, Paul

doi:10.1112/jlms/jdw005

Cited by 6 publications

(8 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In an independent work, Gournay and Jolissaint obtained a formula for the norm of harmonic cocycles [10,Theorem 1.2] which subsume our estimate. Indeed, since the average of B(x, y) over the neighbors y of x is proportional to the indicator function on ∂X, the cocycle B : X 2 → C(∂X)/C is harmonic (1) , i.e.…”

Section: Introductionmentioning

confidence: 53%

“…Therefore P B yields an harmonic 1-cocycle of Aut(X) for its regular representation into ℓ 2 (E). Viewing it as an inhomogeneous 1-cocycle, their result implies Theorem 3 (Gournay-Jolissaint [10,Theorem 1.2]). For every integer q ≥ 2, there are constants C ′ , K ′ > 0 such that P B(x, y) 2 = C ′ d(x, y) − K ′ (1 − q −d(x,y) ), for all x, y ∈ X.…”

Section: Introductionmentioning

confidence: 80%

See 1 more Smart Citation

Norm growth for the Busemann cocycle

Dumont¹

2018

Bull. Belg. Math. Soc. Simon Stevin

View full text Add to dashboard Cite

Using explicit methods, we provide an upper bound to the norm of the Busemann cocycle of a locally finite regular tree X, emphasizing the symmetries of the cocycle. The latter takes value into a submodule of square summable functions on the edges of X, which corresponds the Steinberg representation for rank one groups acting on their Bruhat-Tits tree. The norm of the Busemann cocycle is asymptotically linear with respect to square root of the distance between any two vertices. Independently, Gournay and Jolissaint [10] proved an exact formula for harmonic 1-cocycles covering the present case.

show abstract

Section: Introductionmentioning

confidence: 53%

Section: Introductionmentioning

confidence: 80%

Norm growth for the Busemann cocycle

Dumont¹

2018

Bull. Belg. Math. Soc. Simon Stevin

View full text Add to dashboard Cite

show abstract

“…There are clearly cases outside those described in the previous subsection where virtual coboundaries are useful. In this subsection, the setting is that of [29]; as such the groups may not be finitely generated. These are groups acting on trees by automorphism.…”

Section: Virtual Coboundary For Groups Acting On Treesmentioning

confidence: 99%

“…Note that in this context, there are no constant functions in W ! Let us now describe a virtual coboundary for the Haagerup cocycle (see [29] for further background and see [20, §3] for another possible choice). A simple computations shows that f may be defined as follows:…”

Section: Virtual Coboundary For Groups Acting On Treesmentioning

confidence: 99%

Mixing, malnormal subgroups and cohomology in degree one

Gournay

2018

Groups Geom. Dyn.

View full text Add to dashboard Cite

The aim of the current paper is to explore the implications on the group G of the non-vanishing of the cohomology in degree one of one of its representation π, given some mixing conditions on π. For example, harmonic cocycles of weakly mixing unitary representations factorise by the FC-centre. In that case non-vanishing implies the FC-centre is trivial or fixes a vector. Next, for any subgroup H < G, H will either be "small", almost-malnormal or π |H also has non-trivial cohomology in degree one (in this statement, "small", reduced vs unreduced cohomology and unitary vs generic depend on the mixing condition). The notion of q-normal subgroups is an important ingredient of the proof and results on the vanishing of the reduced ℓ p -cohomology in degree one are obtained as an intermediate step.

show abstract

“…Our other application concerns direct product. Given two graph H 1 = (X 1 , E 1 ) and Proposition 1 and Corollary 3 also have consequences on the cohomology of Hilbertian representations with ℓ p -coefficients, see [7,Corollary 2.6]. The same can be said for some representations given by G L q (with coefficients in ℓ p ) modulo the following remark:…”

Section: Introductionmentioning

confidence: 97%

Harmonic functions with finite p-energy on lamplighter graphs are constant

Gournay

2016

Comptes Rendus. Mathématique

View full text Add to dashboard Cite

The aim of this note is to show that lamplighter graphs where the space graph is infinite and at most two-ended and the lamp graph is at most two-ended do not admit harmonic functions with gradients in ℓ p (i.e. finite p-energy) for any p ∈ [1, ∞[ except constants (and, equivalently, that their reduced ℓ p cohomology is trivial in degree one). This answers a question of Georgakopoulos [3] on functions with finite energy in lamplighter graphs. The proof relies on a theorem of Thomassen [16] on spanning lines in squares of graphs. Using similar arguments, it is also shown that many direct products of graphs (including all direct products of Cayley graphs) do not admit non-constant harmonic function with gradient in ℓ p .

show abstract

Conditionally negative type functions on groups acting on regular trees

Cited by 6 publications

References 14 publications

Norm growth for the Busemann cocycle

Norm growth for the Busemann cocycle

Mixing, malnormal subgroups and cohomology in degree one

Harmonic functions with finite p-energy on lamplighter graphs are constant

Contact Info

Product

Resources

About