Discrete Quasiconformal Groups I

Cyclic Parabolic Quasiconformal Groups That Are Not Quasiconformal Conjugates of M Obius Groups

P. Tukia published in T2] the rst example of a uniformly quasi-isometric and hence quasiconformal group acting on R n , n 3, which is not a quasiconformal conjugate of any MM obius group. We have analyzed this group and have shown that it contains elements which generate cyclic parabolic uniformly quasi-isometric groups that cannot be conjugated by a quasi-conformal map to a MM obius group. By ...

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. Le...

Structural Properties of Inner Amenable Discrete Groups

Conformal and Quasiconformal Realizations of Exceptional Lie Groups

We present a nonlinear realization ofE8(8) on a space of 57 dimensions, which is quasiconformal in the sense that it leaves invariant a suitably defined “light cone” in R 57. This realization, which is related to the Freudenthal triple system associated with the unique exceptional Jordan algebra over the split octonions, contains previous conformal realizations of the lower rank exceptional Lie...

Quasiconformal distortion of projective transformations and discrete conformal maps

We consider the quasiconformal dilatation of projective transformations of the real projective plane. For non-affine transformations, the contour lines of dilatation form a hyperbolic pencil of circles, and these are the only circles that are mapped to circles. We apply this result to analyze the dilatation of the circumcircle preserving piecewise projective interpolation between discretely con...