Bounded geometry for Kleinian groups.

*(English)*Zbl 1061.37026Let \(N\) be a hyperbolic 3-manifold homeomorphic to the interior of a compact manifold. Then \(N\) has bounded geometry if, outside the cusps of \(N,\) there is a lower bound on the length of all closed geodesics in \(N.\) The condition of bounded geometry is very helpful in understanding some basic questions, such as classification by end invariants (Thurston’s ending lamination conjecture), and description of topological structure of the limits set. Bounded geometry also has implications for the spectral theory of a hyperbolic 3-manifold and its \(L^{2}\)-cohomology. The end invariants of \(N\) are points in a certain parameter space associated to each end, which describe the asymptotic geometry of \(N.\)

Given now the case where \(N\) is homeomorphic to \(S \times {\mathbb R},\) where \(S\) is a surface of finite type. This manifold is quotient of \({\mathbb H}^{3}\) by injective representations \(\rho : \pi_{1}(S) \rightarrow PSL_{2}({\mathbb C})\) with discrete images (a marked Kleinian surface group). The theory of Ahlfors-Bers attaches to \(\rho\) two invariants \((\nu_{+}, \nu_{-})\) lying in a combination of Teichmüller spaces and lamination spaces of \(S\) and its subsurfaces. Moreover, the pair \((\nu_{+}, \nu_{-})\) will associated a collection of positive integers \(\{d_{Y}(\nu_{+}, \nu_{-})\},\) where \(Y\) runs over all isotopy classes of essential subsurfaces in \(S.\)

Bounded geometry theorem: Let \(\rho : \pi_{1}(S) \rightarrow PSL_{2}({\mathbb C})\) be a Kleinian surface group with no accidental parabolics, and end invariants \((\nu_{+}, \nu_{-}).\) Then \(\rho\) has bounded geometry if and only if the coefficients \(\{d_{Y}(\nu_{+}, \nu_{-})\}\) are bounded from above. Moreover, for any \(K > 0\) there exists \(\varepsilon > 0,\) depending only on \(K\) and the topological type of \(S,\) so that \(\sup_{Y}d_{Y}(\nu_{+}, \nu_{-}) < K\) implies that \(\inf_{\gamma}l_{\rho}(\gamma) > \varepsilon,\) where the infimum is over elements of \(\pi_{1}(S)\) that are not externally short, and the supremum is over essential subsurfaces for which \((\nu_{+}, \nu_{-})\) is defined.

Similarly, given \(\varepsilon\) there exists \(K\) for which the implication is reversed. Also, the author obtains corollaries on ending lamination for bounded geometry and Bers density for bounded geometry.

Given now the case where \(N\) is homeomorphic to \(S \times {\mathbb R},\) where \(S\) is a surface of finite type. This manifold is quotient of \({\mathbb H}^{3}\) by injective representations \(\rho : \pi_{1}(S) \rightarrow PSL_{2}({\mathbb C})\) with discrete images (a marked Kleinian surface group). The theory of Ahlfors-Bers attaches to \(\rho\) two invariants \((\nu_{+}, \nu_{-})\) lying in a combination of Teichmüller spaces and lamination spaces of \(S\) and its subsurfaces. Moreover, the pair \((\nu_{+}, \nu_{-})\) will associated a collection of positive integers \(\{d_{Y}(\nu_{+}, \nu_{-})\},\) where \(Y\) runs over all isotopy classes of essential subsurfaces in \(S.\)

Bounded geometry theorem: Let \(\rho : \pi_{1}(S) \rightarrow PSL_{2}({\mathbb C})\) be a Kleinian surface group with no accidental parabolics, and end invariants \((\nu_{+}, \nu_{-}).\) Then \(\rho\) has bounded geometry if and only if the coefficients \(\{d_{Y}(\nu_{+}, \nu_{-})\}\) are bounded from above. Moreover, for any \(K > 0\) there exists \(\varepsilon > 0,\) depending only on \(K\) and the topological type of \(S,\) so that \(\sup_{Y}d_{Y}(\nu_{+}, \nu_{-}) < K\) implies that \(\inf_{\gamma}l_{\rho}(\gamma) > \varepsilon,\) where the infimum is over elements of \(\pi_{1}(S)\) that are not externally short, and the supremum is over essential subsurfaces for which \((\nu_{+}, \nu_{-})\) is defined.

Similarly, given \(\varepsilon\) there exists \(K\) for which the implication is reversed. Also, the author obtains corollaries on ending lamination for bounded geometry and Bers density for bounded geometry.

Reviewer: V. V. Chueshev (Kemerovo)

##### MSC:

37F30 | Quasiconformal methods and Teichmüller theory, etc. (dynamical systems) (MSC2010) |

57M50 | General geometric structures on low-dimensional manifolds |

30F40 | Kleinian groups (aspects of compact Riemann surfaces and uniformization) |

30F60 | Teichmüller theory for Riemann surfaces |

32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |