On the degeneration of Schottky groups.
(Sur la dégénérescence des groupes de Schottky.)

*(French)*Zbl 0828.57008Suppose \(\lambda\) and \(\lambda'\) are two measured laminations which fill a hyperbolic surface \(S\) (this means that a lift of any component of \(S- (\lambda \cup \lambda')\) to \(\text{H}^2\) is either bounded or lies in a horoball region of a cusp point). The Double Limit Theorem of Thurston says that given a sequence \((\rho_i : \pi_1 (S) \to \text{PSL} (2,C))\) of quasifuchsian groups, if the corresponding sequences of hyperbolic structures on the quotient surfaces of the two components of the region of discontinuity converge to \(\lambda\) and \(\lambda'\), then the sequence \((\rho_i)\) has a convergent subsequence. Thurston has conjectured an analogue for Schottky groups, i.e. geometrically finite structures on a 3-dimensional handlebody \(H\). Consider pairs \((\rho, \varphi)\), where \(\rho : G = \pi_1 (H) \to \text{PSL} (2,C)\) and \(\varphi\) is a diffeomorphism from \(H\) to \((\text{H}^3 \cup \Omega (\rho (G)))/o(G)\). By the Ahlfors-Bers Theorem the space of such structures is homeomorphic to the Teichmüller space \({\mathcal T} (\partial H)\). Let \(\lambda\) be a geodesic lamination in the Thurston boundary of \({\mathcal T} (\partial H)\), and assume that \(\lambda\) lies in the Masur domain (a certain open subset of \({\mathcal T} (\partial H)\) on which the mapping class group of \(H\) acts properly discontinuously). The conjecture is that given a sequence of geometrically finite structures \(((\rho_i, \varphi_i))\), if the corresponding hyperbolic structures on \(\partial H\) converge to \(\lambda\), then \((\rho_i)\) has a convergent subsequence. In this paper, the author proves Thurston’s conjecture for the case when \(H\) has genus 2 and each component of \(\partial H - \lambda\) is simply-connected. This overlaps partially with cases proven by R. Canary.

The author’s work is in the context of a more general conjecture, motivated as follows. Suppose that \((\rho_i)\) has no convergent subsequence. By a result of J. Morgan and P. Shalen, one can obtain an action of \(G\) on an R-tree \(T\) such that the stabilizer of any noncyclic vertex is cyclic (i.e. “the action has small stabilizers”), and such that for some subsequence of \((\rho_i)\), the ratio of the translation lengths of any two elements on \(G\) (for which the denominator does not have translation length 0) is the limit of the ratios of their translation lengths as elements of the \(\rho_i (G))\). There is a \(G\)- equivariant map \(F\) (well-defined up to \(G\)-equivariant homotopy) from the universal cover \(\text{H}^3\) of \(H\) to \(T\). Let \(P^1 (\text{H}^3)\) be the projective unit tangent bundle of \(\text{H}^3\), and let \(P : P^1 (\text{H}^3) \to \text{H}^3\) be the projection. A closed subset \(X \subset P^1 (\text{H}^3)\), invariant under the geodesic flow, is said to be realized in \(T\) if the restriction of \(F \circ P\) to \(X\) is \(G\)-equivariantly homotopic to a map which carries each geodesic in \(X\) injectively into \(T\). The general conjecture is that for any action of \(G\) on an \(R\)-tree \(T\), with small stabilizers, every measured lamination in the Masur domain is realized in \(T\).

The author proves the conjecture for minimal laminations when the genus of \(H\) is 2 or (more generally) when the action of \(G\) on \(T\) is dual to a measured lamination in a compact surface with boundary. This implies the case of Thurston’s conjecture given above. The proofs use a variety of geometric methods, together with results of R. Skora, and M. Culler and K. Vogtmann.

The author’s work is in the context of a more general conjecture, motivated as follows. Suppose that \((\rho_i)\) has no convergent subsequence. By a result of J. Morgan and P. Shalen, one can obtain an action of \(G\) on an R-tree \(T\) such that the stabilizer of any noncyclic vertex is cyclic (i.e. “the action has small stabilizers”), and such that for some subsequence of \((\rho_i)\), the ratio of the translation lengths of any two elements on \(G\) (for which the denominator does not have translation length 0) is the limit of the ratios of their translation lengths as elements of the \(\rho_i (G))\). There is a \(G\)- equivariant map \(F\) (well-defined up to \(G\)-equivariant homotopy) from the universal cover \(\text{H}^3\) of \(H\) to \(T\). Let \(P^1 (\text{H}^3)\) be the projective unit tangent bundle of \(\text{H}^3\), and let \(P : P^1 (\text{H}^3) \to \text{H}^3\) be the projection. A closed subset \(X \subset P^1 (\text{H}^3)\), invariant under the geodesic flow, is said to be realized in \(T\) if the restriction of \(F \circ P\) to \(X\) is \(G\)-equivariantly homotopic to a map which carries each geodesic in \(X\) injectively into \(T\). The general conjecture is that for any action of \(G\) on an \(R\)-tree \(T\), with small stabilizers, every measured lamination in the Masur domain is realized in \(T\).

The author proves the conjecture for minimal laminations when the genus of \(H\) is 2 or (more generally) when the action of \(G\) on \(T\) is dual to a measured lamination in a compact surface with boundary. This implies the case of Thurston’s conjecture given above. The proofs use a variety of geometric methods, together with results of R. Skora, and M. Culler and K. Vogtmann.

Reviewer: D.McCullough (Norman)

##### MSC:

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

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

30F60 | Teichmüller theory for Riemann surfaces |

57N05 | Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010) |

##### Keywords:

sequence of quasifuchsian groups; handlebody of genus 2; action on an \(R\)-tree; measured lamination; hyperbolic surface; sequences of hyperbolic structures; Schottky groups; geometrically finite structures on a 3-dimensional handlebody; Teichmüller space; Masur domain; small stabilizers; minimal laminations
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