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On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions. (English) Zbl 0567.58015
Riemann surfaces and related topics: Proc. 1978 Stony Brook Conf., Ann. Math. Stud. 97, 465-496 (1981).
[For the entire collection see Zbl 0447.00006.]
Let $$\Gamma$$ be a discrete group of hyperbolic isometries of 3- dimensional hyperbolic space $$H^ 3$$. Then $$\Gamma$$ is also a group of conformal transformations of $$S^ 2$$. Then there is no $$\Gamma$$- invariant measurable vector field on the conservative part of $$S^ 2$$ for the action with respect to Lebesgue measure. In contrast, on the dissipative part of $$S^ 2$$, the space of $$\Gamma$$-invariant line fields is infinite dimensional. A measurable version of the Riemann mapping theorem due to Ahlfors and Bers gives a one-to-one correspondence between measurable vector fields and quasiconformal (q-c) homeomorphisms (up to composition with a conformal homeomorphism). So on the conservative part of $$S^ 2$$ there are no conformal quasiconformal deformations of $$\Gamma$$. A q-c deformation of $$\Gamma$$ is a group $$\Gamma$$ ’ of hyperbolic isometries conjugated to $$\Gamma$$ by a q-c homeomorphism. So if $$\Gamma$$ acts conservatively on $$S^ 2$$, $$\Gamma$$ is rigid - there are no nonconformal q-c deformations, equivalently no nonisometric pseudo-isometric deformations of $$\Gamma$$ acting on $$H^ 3.$$
It is shown that if $$\Gamma$$ has infinite solid angle $$(\Sigma_{\gamma}| \gamma '(z)|^ 2=\infty$$, where $$\gamma$$ ’ denotes the conformal derivative) then not only is $$\Gamma$$ conservative on $$S^ 2$$, but $$\Gamma$$ is ergodic on $$S^ 2\times S^ 2$$; equivalently the geodesic flow for the manifold $$H^ 3/\Gamma$$ is ergodic. These two last conditions are actually equivalent to infinite solid angle, and to almost all points on $$S^ 2$$ having ”conical approach”. A point has horospherical approach if arbitrarily small horospheres based on that point contain elements of a fixed $$\Gamma$$- orbit in $$H^ 3$$. The conservative part of $$S^ 2$$ coincides (up to measure zero) with the set of points with horospherical approach, and the dissipative part coincides (up to measure zero) with the set of points with a minimal horosphere containing an element of the $$\Gamma$$-orbit. It is shown that, besides full measure for horospherical approach, other equivalent conditions for conservativity are that the fundamental domain of $$\Gamma$$ in $$H^ 3$$ have zero area in the boundary $$S^ 2$$, and that $$\lim_{r\to \infty}V(r)/H(r)=0$$, where V(r), H(r) are the volumes of the balls of radius r in $$H^ 3/\Gamma$$, $$H^ 3$$ respectively.
If $$\Gamma$$ is finitely generated, the space of q-c deformations must be finite dimensional. Since a different group is obtained from each $$\Gamma$$-invariant vector field on the limit set of $$\Gamma$$, and the space of these is infinite dimensional if the action of $$\Gamma$$ is partially dissipative on the limit set, $$\Gamma$$ must be conservative on its limit set, and work of Ahlfors implies that the space of q-c deformations is a complex manifold.

MSC:
 37A99 Ergodic theory 53C30 Differential geometry of homogeneous manifolds 57R99 Differential topology 22E40 Discrete subgroups of Lie groups 30C62 Quasiconformal mappings in the complex plane