This is a subset of the space of measures on the space G K of geodesics in H2 intersecting K , with the topology of weak convergence of measures. In this topology, the bending cocycle does not depend continuously on the lamination.

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One reason for this is the behaviour of the laminations near the endpoints of the segment over which we evaluate the cocycle. Then fn g converges weakly to the zero lamination, but Copyright Geometry and Topology Christos Kourouniotis the cocycle of n relative to [ei ; i] is approximately a hyperbolic isometry of translation length 1. Epstein and Marden nd conditions under which a sequence of measured laminations gives a convergent sequence of cocycles relative to a given pair of points.

In this article we show that when the lamination is invariant by a discrete group and we only consider cocycles relative to points in the orbit of a suitable point x 2 H2 , any sequence of measured laminations fn g which converges weakly gives rise to cocycles which converge up to conjugation.

We show further that the same conjugating elements can be used for the cocycles for n corresponding to the dierent generators of the group. Hence the laminations n determine bending homomorphisms which, after conjugation by suitable isometries, converge to the bending homomorphism determined by 0. This implies that the deformations converge in Q S.

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Let fn g be a sequence of complex measured geodesic laminations, converging weakly to a lamination 0. Then the bending deformations Bn : Dn! We also state an innitesimal version of the Theorem. These results do not necessarily imply the continuous dependence of the deformation on the bending lamination, because the space of measured laminations is not rst countable.

If however we restrict our attention to the subset of measured laminations with non negative real and imaginary parts, then we can apply results in [6] to obtain the following Theorem. T [] is continuous, and holomorphic in []. Geometry and Topology Monographs, Volume 1 On the continuity of bending The proof of Theorem 1 is based on the observation that, when the lamination is invariant by a discrete group and we are considering cocycles with respect to points x and g x , for some g in the group, the eect of a lamination near the endpoints of the segment [x; g x ] is controlled by its eect near x, provided that the lamination does not contain geodesics very close to the geodesic carrying [x; g x ].

This last condition can be achieved by choosing x to be a point not on the axis of a conjugate of g see Corollary 2. In Section 1 we describe the space of measured laminations and we recall the denition of bending. In the beginning of Section 2 we recall or modify certain results from [2] and [5] which provide bounds for the eect of bending along nearby geodesics. Lemma 2. The proof of Theorems 1, 2 and 3 is given in Section 3. The laminations n are replaced by nite approximations.

The main result is Lemma 3. Then a diagonal argument is used to obtain the convergence of bending. We x a hyperbolic structure on S , and let 0 : 1 S! P SL 2; C acts on the left on R by inner automorphisms. We denote the equivalence class in Q S of a homomorb left phism 2 R by []. P SL 2; C obtained by conjugation with the quasiconformal homeomorphism b! We want to study the deformation of quasi-Fuchsian structures by bending, [4], [2], [5]. Bending is determined by a geodesic lamination on S with a complex valued transverse measure. A measured geodesic lamination on S lifts to a measured geodesic lamination on H2.

The set G K of geodesics in H2 intersecting K is a compact metrizable space. The set of measured geodesic laminations on S can be considered as a subset of M G K , the set of complex valued Borel measures on G K.

A measured geodesic lamination on S is called nite if it is supported on a nite set of simple closed geodesics in S. Then, for any compact subset K of H2 , the measure on G K determined by the lift of to H2 has nite support. Given a nite measured geodesic lamination on S , we dene bending the quasi-Fuchsian structure [] on S as follows. Choose a point x on H2 and, for each gj , consider the geodesic segment [x; gj x ]. In such cases either the choice of the lift will not matter, or there will be an obvious choice.

Any measured geodesic lamination on S can be approximated by nite laminations so that the corresponding bending deformations converge, [2], [5]. Then the following are equivalent.

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Let be a maximal geodesic lamination on S , and : S! H3 be the lift of. There is a positive number N with the following property. Then there is a positive number N with the following property. This problem falls naturally into two cases. A The two hyperbolic elements have intersecting axes. See for instance [4] where this is shown by explicitly exhibiting two pairs of geometric circles, one paired by A and one by B. In this case the quotient surface is a one holed torus and, as any generating pair will have intersecting axes, we see that G is classical on every possible generating pair.

B The hyperbolic elements have non-intersecting axes. If so then all generating pairs of G must have non-intersecting axes, or else we are back in case A. First suppose G is a classical Schottky group on these two generators A and B. Without loss of generality we can replace any generator by its inverse so that we get a picture such as the one in gure 5, with the arrows on the two generators in the same direction.

The quotient surface is a three holed sphere. Theorem 2 A group G that has a quotient surface which is not a one holed Geometry and Topology Monographs, Volume 1 Jack Button torus cannot be classical on all generating sets. Proof We have already considered any G generated by two elements. Given any G generated by three or more elements, we can nd a pair of generators with non-intersecting axes, and use the above argument on the subgroup generated by this pair.

As the subgroup is not classical on all generating sets, nor is G.

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## Hilbertâ€™s 3rd Problem and Invariants of 3{manifolds Geometry & Topology Monographs 383

Finally we show the existence of a Fuchsian group generated by two elements which is Schottky, but not classical, on this generating pair. Proof If we know G is classical on hA; Bi then we can build up a pattern of nested circles as in gure 5, and see the location of the xed points of the axes directly. Conversely if we only have information as in gure 6 then we consider the image of a suitable point x under the generators. We can take any two points y and z in the interval between Ax and Bx, and use these as endpoints for the geometric circles we require.

This provides us with our two pairs of circles which show that G is discrete, and classical on hA; Bi.

Theorem 3 The Fuchsian group in gure 7, which is Schottky on the generators A and B , is not classical on them. Analyse Math. Related documents. Electrical characterization of semiconductor materials and. Classical Period. Principles of Inorganic Chemistry. Download advertisement. State Flags. Probability Theory.. Probability, random variables, stochastic processes 3ed. Quantum Groups and Knot Algebra s T. Quantum groups and their representations Springer, T s.

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