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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.

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 |