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The end result is then an isotopy, deforming h(x) to the identity for x3 = 0. Since the construction did not move points with | JC3 | ^ i , we can make the same deformation simultaneously at all planes x3 = m, meZ, and obtain the identity on the planes. We then consider in the same way first x2 = rri and then xY = m\ with m\ m" e Z, and obtain the identity on these planes, without moving points in x3 = m or, in the second step, in x2 = m'. The end result is an isotopy h(x, t) such that h(x, 0) = h(x) and |dh/dt\ ^ C, where H(x) = h(x, 1) = x on dQ for all unit cubes Q associated with the lattice L0 .

Nauk SSSR 20 (1938), 241-242. 5. M. A. Lavrent'ev and P. P. Belinskii, Some problems of the geometric theory of functions. Jubilee volume in honor of 80th birthday of Academician I. M. Vinogradov, Trudy Mat. Inst. Steklov 2, (1972). 6. O. Martio, S. Rickman, and J. Väisälä, Topological and metric properties of quasiregular mappings, Ann. Acad. Sei. Fenn. Ser. AI 488, (1971), 1-32. 7. V. A. Zorich, Lavrent'ev's theorem on quasiconformal mappings in space, Mat. Sb. 74 (1967), 417-433. A Theorem on the Boundary Correspondence under Conformai Mapping with Application to Free Boundary Problems of Fluid Dynamics PAUL W.

The following theorem has been proved by Nielsen [13]. Theorem (Nielsen). If G is a noncommutative subgroup of PU (1,1) which contains only hyperbolic transformations (besides the identity), then G is discrete. This theorem has been generalized in several ways. Fenchel and Nielsen [5] and Siegel [17] have proved the following. ) Theorem (Fenchel-Nielsen-Siegel). If G is a subgroup of PU(\, 1) such that (a) the elements of G do not leave invariant a point or pair of points on the boundary dD, and (b) the identity is not an accumulation point of the elliptic elements in G, then G is discrete.

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