Filling invariants of stratified nilpotent Lie groups

Abstract

Filling invariants are measurements of a metric space describing the behaviour of isoperimetric inequalities. In this article we examine filling functions and higher divergence functions. We prove for a class of stratified nilpotent Lie groups that in the low dimensions the filling functions grow as fast as the ones of the Euclidean space and in the high dimensions slower than the filling functions of the Euclidean space. We do this by developing a purely algebraic condition on the Lie algebra of a stratified nilpotent Lie group. Further, we find a sufficient criterion for such groups to have a filling function in a special dimension with faster growth as the appropriate filling function of the Euclidean space . Further we bound the higher divergence functions of stratified nilpotent Lie groups. arXiv:1507.04871v3 [math.DG] 3 Feb 2017where the infimum is taken over all (m + 1)-chains b with ∂b = a and the supremum is taken over all m-cycles a with mass(a) ≤ l.As our main interest lies in the large scale geometry of the space X, the exact description of the filling functions is of less importance to us. Indeed we only look at the asymptotic growth rate of this functions. We do this by the following equivalence relation, which makes the growth rate of the filling functions an quasi-isometry invariant.Definition. Let f, g : R + → R + be functions. Then we write f g if there is a constant C > 0 with f (l) ≤ Cg(Cl) + Cl + C ∀l ∈ R + .If f g and g f we write f ∼ g. This defines an equivalence relation.We read this notation F H as "F is bounded from above by H" respectively "H is bounded from below by F " depending on whether we are more interested in F or H.A special case is that of a positive 2-parameter m-family F = {f s,t } s,t bounded from above (or below) by a constant positive 2-parameter m-family H, i.e. H = {h} s,t . This means that all functions f s,t areWe denote the set of m-dimensional integral currents on a complete metric space X by I m (X).Definition. Let X be a complete metric space and let m ∈ N. Then X satisfies an isoperimetric inequality of rank δ for I m (X), if there is a constant C > 0, such that for every integral current T ∈ I m (X) with ∂T = 0, there exists an integral current S ∈ I m+1 (X) with ∂S = T and M (S) ≤ C · M (T ) δ . Stratified nilpotent Lie groupsA Lie group G with Lie algebra g is called nilpotent, if its lower central series

Higher divergence for nilpotent Lie groups

“…We prove that R ψ(η) (a) is contained in the D-neighbourhood of a for D : = c φ +Lip(ψ)·(Lip(g −1 )+ D η + diam(W η,ε )). Then the claim follows by Lemma 2.4 and equation (4).…”

“…So we obtain ∂ R i (a) (4) = s 2 i ( ∂R ψ(η) (s 2 −i (a))) = s 2 i ( P 1 (s 2 −i (a)) − P 0 (s 2 −i (a))) (2) = s 2 i (s 2 ( P 0 (s 2 −1 (s 2 −i (a))))) − s 2 i ( P 0 (s 2 −i (a)))…”

“…As the complex Heisenberg group H n C is k-approximable for k ≤ n (see [11]), the Main Theorem combined with the results on the higher divergence from [4] yields:…”

“…Proof. Using the Main Theorem, the growth of the k-dimensional divergence follows by [11,Corollary 4.14] for k < n, by [11,Theorem 8] for k = n and by [4,Corrollary 5.4] for k > n.…”

“…Using the results of [4] and [5], we get the corresponding behaviour for the divergence of the quaternionic and octonionic Heisenberg groups H n H and H n O in the dimensions ≤ n and codimensions < n. Only for the octonionic Heisenberg groups H n O we haven't the lower bound for the divergence in dimension n, because of the missing lower bound for the filling function in this dimension. Then one approximates B in the grid τ 1 by B 1 = a 1 − b 1 .…”