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Averages are invariants defined on the ℓ 1 cohomology of Lie groups. We prove that they vanish for abelian and Heisenberg groups. This result completes work by other authors and allows to show that the ℓ 1 cohomology vanishes in these cases.R n ∂u ∂x n dx 1 · · · dx n = 0.A similar argument applies to other coordinates. Note that a n dx 1 ∧ · · · ∧ dx n = (−1) n−1 ω ∧ (dx 1 ∧ · · · ∧ dx n−1 ). More generally, if G is a Lie group of dimension n, there is a pairing, the average pairing, between closed L 1 k-forms ω and closed left-invariant (n − k)-forms β, defined byThe integral vanishes if either ω = dφ where φ ∈ L 1 , or β = dα where α is left-invariant. Indeed, Stokes formula M dγ = 0 holds for every complete Riemannian manifold M and every L 1 form γ such that dγ ∈ L 1 . Hence the pairing descends to quotients, the L 1,1 -cohomology L 1,1 H k (G) = closed L 1 k-forms/d(L 1 (k − 1)-forms with differential in L 1 ), and the Lie algebra cohomology1.2. ℓ q,1 cohomology. It turns out that L 1,1 -cohomology has a topological content. By definition, the ℓ q,p cohomology of a bounded geometry Riemannian manifold is the ℓ q,p cohomology of every bounded geometry simplicial complex quasiisometric to it. For instance, of a bounded geometry triangulation. Contractible Lie groups are examples of bounded geometry Riemannian manifolds for which L 1,1 -cohomology is isomorphic to ℓ 1,1 -cohomology.We do not need define the ℓ q,p cohomology of simplicial complexes here, since, according to Theorem 3.3 of [6], every ℓ q,p cohomology class of a contractible Lie group can be represented by a form ω which belongs to L p as well as an arbitrary finite number of its derivatives. If the class vanishes, then there exists a primitive φ of ω which belongs to L q as well as an arbitrary finite number of its derivatives. This holds for all 1 ≤ p ≤ q ≤ ∞.Although ℓ p with p > 1, and especially ℓ 2 cohomology of Lie groups has been computed and used for large families of Lie groups, very little is known about ℓ 1 cohomology.

Stratified groups are those simply connected Lie groups whose Lie algebras admit a derivation for which the eigenspace with eigenvalue 1 is Lie generating. When a stratified group is equipped with a left-invariant path distance that is homogeneous with respect to the automorphisms induced by the derivation, this metric space is known as Carnot group. Carnot groups appear in several mathematical contexts. To understand their algebraic structure, it is useful to study some examples explicitly. In this work, we provide a list of low-dimensional stratified groups, express their Lie product, and present a basis of left-invariant vector fields, together with their respective left-invariant 1-forms, a basis of right-invariant vector fields, and some other properties. We exhibit all stratified groups in dimension up to 7 and also study some free-nilpotent groups in dimension up to 14. Contents(12457H) 136 (12457L) 138 (12457L 1 ) 140 (123457A) 143 6. Some free-nilpotent groups in low dimension 145 F 23 . 145 F 24 . 146 F 33 . 148 F 25 . 153 References 161

We present a constructive approach to torsion-free gradings of Lie algebras. Our main result is the computation of a maximal grading. Given a Lie algebra, using its maximal grading we enumerate all of its torsion-free gradings as well as its positive gradings. As applications, we classify gradings in low dimension, we consider the enumeration of Heintze groups, and we give methods to find bounds for non-vanishing ℓ q,p cohomology. Contents 1. Introduction 1.1. Overview 1.2. Main results 1.3. Structure of the paper 2. Gradings 2.1. Gradings and equivalences 2.2. Universal gradings 2.3. Gradings induced by tori 2.4. Maximal gradings 2.5. Enumeration of torsion-free gradings 3. Constructions 3.1. Stratifications 3.2. Positive gradings 3.3. Maximal gradings 4. Applications 4.1. Structure from maximal gradings 4.2. Classification of gradings in low dimension 4.3. Enumerating Heintze groups 4.4. Bounds for non-vanishing ℓ q,p cohomology Appendix A. Existence of a positive realization References

In this paper, we study the geometry of surfaces with the generalised simple lift property. This work generalises previous results by Bernstein and Tinaglia [1] and it is motivated by the fact that leaves of a minimal lamination obtained as a limit of a sequence of properly embedded minimal disks satisfy the generalised simple lift property. arXiv:1612.04569v1 [math.GT]