Let Γ be an amenable group and V be a finite dimensional vector space. Gromov pointed out that the von Neumann dimension of linear subspaces of ℓ 2 (Γ; V ) (with respect to Γ) can be obtained by looking at a growth factor for a dynamical (pseudo-)distance. This dynamical point of view (reminiscent of metric entropy) does not requires a Hilbertian structure. It is used in this article to associate to a Γ-invariant linear subspaces Y of ℓ p (Γ; V ) a real positive number dim ℓ p Y (which is the von Neumann dimension when p = 2). By analogy with von Neumann dimension, the properties of this quantity are explored to conclude that there can be no injective Γ-equivariant linear map of finite-type fromHence, we are looking for a notion of dimension for such subspaces, which would increase under injective equivariant linear maps. Inspired by an argument of [6, §1.12] (and partially answering the question found therein), we shall introduce a quantity dim ℓ p which, when p = 2, coincides with definition of von Neumann dimension. This quantity is obtained by a process similar to that of metric entropy or mean dimension, i.e. by looking at an asymptotic growth factor. The definition relies a priori on an exhaustion of Γ, but a generalization of the Ornstein-Weiss lemma in section 5 implies the result is independent of this choice.Though we prove many properties of dim ℓ p , important properties are still lacking. Nevertheless, the results obtained in this paper suffice to establish a non existence result for maps of finite type. We recall their construction.Let D ⊂ Γ be a finite set and let g : V D → V ′ be a continuous map. This data enables the definition of a Γ-equivariant continuous map gRemark that what we denote here as ℓ p (Γ; V ) is more frequently written ℓ p (Γ) ⊗ V .Theorem 1.1. Let Γ be an amenable discrete group. Let V and V ′ be finite dimensional vector spaces. If f :Consequently, if we restrict ourselves to maps of finite type, the question above has a positive answer: there is a Γ-isomorphism of finite type between ℓ p (Γ; R n ) and ℓ p (Γ; R m ) if and only if m = n.

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).

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Lower bound on the equivariant Hilbertian compression exponent α are obtained using random walks. More precisely, if the probability of return of the simple random walk is expThis motivates the study of further relations between return probability, speed, entropy and volume growth. For example, ifUnder a strong assumption on the off-diagonal decay of the heat kernel, the lower bound on compression improves to α ≥ 1 − γ. Using a result from Naor & Peres [27] on compression and the speed of random walks, this yields very promising bounds on speed and implies the Liouville property if γ < 1/2.