** Riv.Mat.Univ.Parma (4) 17 (1991) **

**ALEXANDER G. REZNIKOV**

*The space of spheres and conformal geometry*

In memory of G. I. Katz

**Pages:** 111-130

**Received:** 18 September 1990

**Abstract**
We show that: (1) the complement of the limit set of a Kleinian group acting in *S*^{n} carries an
invariant Finsler metric; (2) every conformal map of a compact domain in *S*^{n} in itself is a
contraction of some Finsler metric given by an explicit formula; (3) for a length-parametrized smooth
curve *γ(t)* in ℜ^{2} the integral
is a conformal invariant (*K* is the curvature), which is an
"integrated" version of a conformal invariance of the Virasoro cocycle and Schwartzian; (4) every non-convex near
infinity hypersurf in half-space of ℜ^{n}
should yield strong curvature decaying conditions; (5) natural
representation of *O*(*n* + 1, 1) in ℜ^{n+1,1} is "glued" to some isometrical Banach representation
(unitar when *n*=2), which is a sort of quantization of "gluing" the conformal action on *S*^{n} to
isometrical action in the hyperbolic space *H*^{n}+1.