The Conformal Modulus And The Moduli Space On The Heisenberg Group

The rank 1 symmetric spaces of non-compact type are real hyperbolic space,complex hyperbolic space,quaternionic hyperbolic space and Cayley hyperbolic plane.There are representations from the Heisenberg manifolds to the rank 1 symmetric space.We can identify the one point compactification of Heisenberg groups with the boundary of the hyperbolic spaces.This thesis involves several problems on the Heisenberg manifold,but we shall mainly focus on the conformal modulus and the moduli space of the points on the Heisenberg group.We summary those as follows:First,we shall consider the conformal modulus of Heisenberg group in the complex space.The Koranyi ellipsoidal ring ε=εB,A,0<B<A,is defined as the image of the Koranyi spherical ring centred at the origin and of radii B and A via a linear contact quasiconformal map L in the Heisenberg group.If K≥1 is the maximal distortion of L then we prove that the modulus of ε is equal tomod(ε)(?).Secondly,we find that the linear contact quasiconformal map L which maps the Koranyi spherical ring to the Koranyi ellipsoidal ring is not the minimiser for the mean distortion.Thirdly,we shall investigate the moduli space of m-tuple of the distinct points in the quaternionic Heisenberg group.Since the one point compactification quater-nionic Heisenberg group can be identified with the boundary of the quaternionic space,we can directly discuss the moduli space of m-tuple of the distinct points in the boundary of the quaternionic space.We deduce then the results to the quaternionic Heisenberg group.Let F1(n,m)be the PSp(n,1)-configuration space of ordered m-tuple of pairwise distinct points in the boundary of quaternionic hyperbolic n-space(?)i.e.,the m-tuple of pairwise distinct points in(?)up to the diagonal action of PSp(n,1).In terms of Cartan’s angular invariant and cross-ratio invariants,the moduli space of F1(n,m)is described by using Moore’s determinant.We show that the moduli space of F1(n,m)is a realdimensional subset of the algebraic variety whose real dimension is the same as F1(n,m)when m>n+1.