Let ~ be an open connected domain in R n, n ~2. If a is a multi-index, a= (ai, ~2 ..... a~)EZ~, the length of a, denoted by [a I, is the integer Xj%~ ~ and D a= (~/~xl) ~" ... (~/~x~)~% A locMly integrable function / on/9 has a weak derivative of order if there is a locally integrable function (denoted by D~ such that fv/( D:cf)dx = (-1) I~j f(D:'/)cfdx for all C ~ functions ~ with compact support in ~. For 1

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Let G be a non-elementary, finitely generated Kleinian group, Λ(G) its limit set and Ω(G) = C\Λ(G) its set of discontinuity. Let δ(G) be the critical exponent for the Poincaré series and let Λ c be the conical limit set of G. We prove that 1.6. The Minkowski dimension of Λ equals the Hausdorff dimension. 7. If area(Λ) = 0 then δ(G) = dim(Λ(G)).

We construct a conformally invariant random family of closed curves in the plane by welding of random homeomorphisms of the unit circle. The homeomorphism is constructed using the exponential of βX where X is the restriction of the two dimensional free field on the circle and the parameter β is in the "high temperature" regime β < √ 2. The welding problem is solved by studying a non-uniformly elliptic Beltrami equation with a random complex dilatation. For the existence a method of Lehto is used. This requires sharp probabilistic estimates to control conformal moduli of annuli and they are proven by decomposing the free field as a sum of independent fixed scale fields and controlling the correlations of the complex dilation restricted to dyadic cells of various scales. For uniqueness we invoke a result by Jones and Smirnov on conformal removability of Hölder curves. We conjecture that our curves are locally related to SLE(κ) for κ < 4. Supported by the Academy of Finland