NON-ELEMENTARY K-QUASICONFORMAL GROUPS ARE LIE GROUPS

Abstract. Suppose that Ω is a subdomain of Rn andG is a non-elementary K-quasiconformal group. Then G is a Lie group acting on Ω. Hilbert-Smith Conjecture states that every locally compact topological group acting effectively on a connected manifold must be a Lie group. Recently Martin [8] has solved the solution of the Hilbert-Smith Conjecture in the quasiconformal category (Theorem 1.2): Theorem 1. Let G be a locally compact group acting effectively by quasiconfor-mal homeomorphisms on a Riemannian manifold. Then G is a Lie group. We will apply the Martin’s theorem in this paper to show the following the-orem. Theorem 2. Suppose that Ω is a subdomain of Rn and G is a non-elementary K-quasiconformal group. Then G is a Lie group acting on Ω. Let Ω and Ω ′ be domains in Rn, n ≥ 2. A homeomorphism f: Ω → Ω ′ is called to beK-quasiconformal if f ∈W 1,nloc (Ω,Rn), the Sobolev space of functions whose first derivatives are locally Ln integrable, and for some K <∞, f satisfies the differential inequality (1) |Df(x)|n ≤ KJ(x, f) almost everywhere in Ω. Here Df(x) is the derivative of f, |Df(x) | is operator norm and J(x, f) is the Jacobian determinant. We say f is quasiconformal if f is K-quasiconformal for some finite K. Thus, quasiconformal homeomorphisms are transformations which have uniformly bounded distortion. They provide a class of mapping