Notes on Groups with Exponential Dehn Function (this Is Not a Preprint)

Abstract

Sketch of proof. First, we use the classical fact that every connected Lie group is quasi-isometric to a simply connected solvable Lie group (when G is algebraic, start from a Levi decomposition G = SU with S reductive and U the unipotent radical, take a minimal parabolic T in S to get an amenable real algebraic group. Decompose the latter as DKV with V the unipotent radical (possibly bigger that U), D a maximal split torus and K a maximal anisotropic torus: then the unit component of DV , in the real topology, is cocompact in G and simply connected solvable). Second, we use that every simply connected solvable Lie group can be described as (R ×R, ∗) with the law of the form (u1, v1) ∗ (u2, v2) = (u1 + u2, P (u1, u2, v1, v2)), where P is a function each component of which, if we denote by (Ui) the 2k coordinates of (u1, u2) and by (Vj) the 2` coordinates of (v1, v2), can be described as a real-valued polynomial in the variables Ui, Vj, and e λkUi , for some finite family of complex numbers λk. For instance, the law of SOL(R) can be described as (u1, x1, y1) ∗ (u2, x2, y2) = (u1 + u2, ex1 + x2, e2y1 + y2) (here (k, `) = (1, 2)). It follows that the n-ball in G (with respect to a compact generating set, e.g. a compact neighbourhood of the identity) is contained in a product B1 × B2, where B1 is a Euclidean ball of linear radius in R, and B2 is a Euclidean ball of exponential radius in R. Let γ be a loop of length ≤ n in G. Then (translating if necessary), γ is contained in B1 × B2. Consider a disc D of area ≤ n, contained in B1 × B2, whose boundary is γ. We have to estimate the area of D in G, i.e. when R is endowed with a ∗-left-invariant Riemannian metric. If x0 ∈ B1 × B2, then the differential at 0 of the left multiplication Lx0 : R k+` → R by x0 has at

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