Non-elementary K-quasiconformal Groups Are Lie Groups

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. 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 quasiconformal 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 theorem. 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 R, n ≥ 2. A homeomorphism f : Ω → Ω′ is called to be K-quasiconformal if f ∈ W 1,n loc (Ω,R), the Sobolev space of functions whose first derivatives are locally L 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 mappings 2000 Mathematics Subject Classification. 30C60.