Boundary values, random walks, and $\ell^p$-cohomology in degree one

Abstract

The vanishing of reduced ℓ 2 -cohomology for amenable groups can be traced to the work of Cheeger & Gromov in [10]. The subject matter here is reduced ℓ p -cohomology for p ∈]1, ∞[, particularly its vanishing. Results for the triviality of ℓ p H 1 (G) are obtained, for example: when p ∈]1, 2] and G is amenable; when p ∈]1, ∞[ and G is Liouville (e.g. of intermediate growth). This is done by answering a question of Pansu in [34, §1.9] for graphs satisfying certain isoperimetric profile. Namely, the triviality of the reduced ℓ p -cohomology is equivalent to the absence of non-constant harmonic functions with gradient in ℓ q (q depends on the profile). In particular, one reduces questions of non-linear analysis (p-harmonic functions) to linear ones (harmonic functions with a very restrictive growth condition).

Mixing, malnormal subgroups and cohomology in degree one

“…It seems natural to introduce the space of p-Dirichlet functions associated to the representation. These function have been of great use to study the cohomology of the left-regular representation, see [25], Martin & Valette [41, §3], Puls [52,53] or §5 below. For S G some finite generating set of G, let…”

“…Hence Proof. Define a transport pattern (see also [25,Definition 3.2] or [24, Definition 4.1]) from φ to ξ (two finitely supported measures) to be a finitely supported function on the edges τ so that ∇ * τ = ξ − φ. The proof consists in showing the equality for any h(·) which is µ-harmonic and has "0-mean"-gradient.…”

“…A very nice application of reduced ℓ p -cohomology in degree one to questions of sphere packings may be found in Benjamini & Schramm [6]. Other important applications include problems of quasi-isometries (see Pansu [49]), the conformal boundary of hyperbolic spaces (see Bourdon & Pajot [12] and Bourdon & Kleiner [11]), the critical exponent for some actions (see Bourdon,Martin & Valette [13]), nonlinear potential theory (see Puls [55] and Troyanov [59]) and existence of harmonic functions with gradient conditions (see [25,Theorem 1.2 or Corollary 3.14] or [28]).…”

“…The first (and shortest) step in the proof of Theorem 1.2.2 (Corollary 5.1) is to show that harmonic cocycles for such representations gives rise to a harmonic function with a gradient in ℓ p which in turn implies that the reduced ℓ p -cohomology in degree one is non-trivial (by a result of [25,Theorem 1.2 or Corollary 3.14]). The second step is to extend the groups for which the ℓ p -cohomology vanishes (see Remark 5.2 for references).…”

Vanishing of ℓp-cohomology and transportation cost

“…Note that the same method used for Theorem 1.4 may be applied to recover this vanishing result, see Theorem 3.2. In [7], the author developed another method to show (among other results) that groups with trivial Poisson boundary (for an SRW on the Cayley graph, that is, Liouville) and superpolynomial growth have trivial reduced p -cohomology (see [ The methods presented here may be of use in non-amenable groups, but apart from groups with infinitely many finite conjugacy classes, the author could not find any other case. They also apply to graphs, but the proper conditions on graphs are not so convenient to formulate (for example, quasi-transitive action by quasi-isometries).…”