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MATHEMATICS RESEARCH CENTRE
Kleinian Groups and
Hyperbolic 3-Manifolds

11-15 September 2001

Organisers: Young Eun Choi, Yohei Komori, Makoto Sakuma, Caroline Series.

An intensive short workshop: 6 lectures by Yair Minsky (Stony Brook)
combinatorial & geometrical aspects of hyperbolic 3-manifolds.

PROGRAMME

Tuesday
11th September
Wednesday
12th September
Thursday
13th September
Friday
14th September
Saturday
15th September
1.30-2.30 Maclachlan 9.15-10.15 Earle 9.15-10.15 Miyachi 9.15-10.15 Bromberg 9.15-10.15 Sakuma
2.45-3.45 Anderson 10.45-11.45 Minsky 10.45-11.45 Minsky 10.45-11.45 Minsky 10.45-11.45 Parker
4.30-5.30 Mednykh 12.00-1.00 Marden 12.00-1.00 Komori 12.00-1.00 Brock 12.00-1.00 Bowditch
2.30-3.30 Minsky 2.30-3.30 Minsky 2.30-3.30 Minsky
4.30-5.30 Markovic 4.30-5.30 Otal 4.30-5.30 Kerckhoff
6.00 Buffet Dinner

Y. Minsky (Stony Brook) Title: Combinatorial & geometrical aspects of hyperbolic 3-manifolds

Abstract:
I will discuss the following basic question: How do we obtain geometric information about a hyperbolic 3-manifold from the asymptotic geometry of its ends? In particular, to a hyperbolic 3-manifold N homotopy-equivalent to a surface S, Thurston and Bonahon associate two "end invariants", which can be conformal structures on S (the classical Ahlfors-Bers construction) or laminations on S. When they are conformal structures they uniquely determine N, and Thurston's "Ending Lamination Conjecture" asserts this is the case also when one or both are laminations. In either case it is hard to give concrete answers to questions such as "What is the injectivity radius of N?", "What is the hyperbolic length of a given curve in N", or "Which geodesics have a length below a given constant?" in terms only of the end invariants.

In these lectures I will outline a mechanism for answering questions like these. From the end invariants, and using the Complex of Curves of S (an interesting simplicial complex that encodes the combinatorial structure of the set of simple curves in S) I will construct a "model manifold" M, which admits a proper Lipschitz homotopy-equivalence to N, and encodes additional information about the location of Margulis tubes in N. If this map could be improved to bi-Lipschitz, it would give a positive answer to Thurston's conjecture; I will discuss such issues at the end, if time permits.


J. Anderson (Southampton) Title: Variations on a theme of Horowitz

Abstract: Horowitz (ref below) showed that for every n greater than or equal to 2, there exist words w_1,..., w_n in F=F(a,b), the free group on two generators, which generate non-conjugate maximal cyclic subgroups of F with the property that Trace(\rho(w_1)) = ... = Trace(\rho(w_n)) for all faithful representations \rho of F into SL(2,C). Randol (ref below) used this result to show that the length spectrum of a hyperbolic surface has unbounded mulitplicity. Masters (ref below) has recently extended this to hyperbolic 3-manifolds. The purpose of this note is to present what is known about traces of faithful representations of F into SL(2,C), and to give a topological characterization of such n-tuples of elements for faithful representations of a closed surface group G.

R. D. Horowitz, `Characters of Free Groups Represented in the Two-Dimensional Special Linear Group', Comm. Pure Appl. Math. 25 (1972), 635--649.

J. D. Masters, `Length multiplicities of hyperbolic 3-manifolds', Israel J. Math. 119 (2000), 9--28.

B. Randol, `The length spectrum of a Riemann surface is always of unbounded multiplicity', Proceedings A. M. S. 78 (1980), 455--456.


B. Bowditch (Southampton), Title: The Cannon-Thurston map for punctured surface bundles

Abstract: Let M be a hyperbolic 3-manifold fibring over the circle. Let G be fibre subgroup, acting on hyperbolic 3-space. We can also realise G as a finite area hyperbolic surface. In the compact case, Cannon and Thurston constructed a G-equivariant map between the boundaries S^1 and S^2. A generalisation of Minsky shows that all one requires is an action of G on hyperbolic 3-space such that the quotient has positive injectivity radius. Another approach was given by Mitra. We extend this to the finite-area case. For the generalisation, we demand that the quotient has positive injectivity radius away from the cusps. Special cases of this map for once-punctured torus bundles were studied by Alperin, Dicks and Porti, and by Dicks and Cannon. Their results suggest some interesting further questions about this map.


J. Brock (Chicago) Title: Pleated surfaces, geometric finiteness, and simultaneous grafting in rigid surface groups

Abstract: I will discuss joint work with K. Bromberg exhibiting a simultaneous grafting procedure in doubly degenerate hyperbolic 3-manifolds with two thin ends. Using Thurston's pleated surfaces and results of Bowditch, we show such simultaneous graftings are geometrically finite cone-manifolds. By simultaneously grafting along pairs of short curves exiting each end, we produce quasi-Fuchsian cone-manifolds approximating the original doubly degenerate manifold. We will conclude by describing how together with Minsky's bounded geometry theorem the grafting procedure produces an approach to Sullivan and Thurston's strengthening of Bers' density conjecture for general surface groups.


K. Bromberg (CalTech) Title: Projective structures with degenerate holonomy

Abstract: We will construct a family of projective structures with the holonomy of a singly degenerate Kleinian surface group. These projective structures can be used to construct a family of quasi-fuchsian hyperbolic cone-manifolds. We will also explain how these cone-manifolds give an approach to the Bers' density conjecture.


C. Earle (Cornell) Title: "Hyperbolic metrics" on complex manifolds

Abstract: A remarkable property of the hyperbolic metric on the unit disk in the plane is that all holomorphic maps of the disk into itself are Lipschitz continuous in that metric, with Lipschitz constant one. There are various systems for assigning pseudometrics to complex manifolds so that holomorphic mappings between them have this Lipschitz property. We shall describe one or two of them and discuss applications to Teichmueller spaces.


S. Kerckhoff (Stanford) Title: Universal bounds on hyperbolic Dehn surgery

Abstract: This will be a discussion of recent joint work with C. Hodgson which provides a bound on the number of non-hyperbolic fillings on a finite volume hyperbolic 3-manifold. There are also strong, universal connections between geometric
quantities, like the volume, of a cupsed manifold and the resulting filled ones.


Y. Komori (Osaka City) Title: The linear slice of the quasifuchsian space of punctured tori

Abstract: For a marked quasifuchsian punctured torus group G=(V, W), the complex length l_V and the complex twist tau_{V,W} define a holomorphic embedding of the quasifuchsian space QF of a punctured torus into the complex plane C^2. It is called the complex Fenchel Nielsen coordinates of QF. For a positive real value c, let Q_{V, c} be the linear subspace of C^2 defined by the condition that l_V=c. Then we can define the linear slice L_c of QF as the intersection of QF and Q_{V, c} which is a holomorphic slice of QF. L_c contains a unique component BM_c intersecting the Fuchsian locus F of QF, studied by Keen-Series, McMullen and Parker-Parkkonen. In this talk, geometric properties of BM_c, especially those of the boundary of BM_c will be shown. We also show that if c is sufficiently small, then L_c=BM_c whereas c is sufficiently large, L_c has other components. This is joint work with Yasushi Yamashita (Nara Women's Univ.).


C. Maclachlan (Aberdeen) Title: Two-elliptic generator arithmetic kleinian groups.

Abstract: Finite covolume Kleinian groups with two elliptic generators arise in a number of situations - some tetrahedral groups, small covolume groups, generalised triangle groups, some Dehn surgeries on two bridge link complements. In a number of critical examples, these turn out to be arithmetic. In joint work with G. Martin, the number of conjugacy classes of such groups has been shown to be finite and some special classes, e.g. noncocompact, have been enumerated. In this talk I will discuss these results and progress towards complete classification.


A. Marden (Minnesota) Title: Simply Connected Regions: The View From Above

Abstract: This will be an introduction to recent work with David Epstein and Vlad Markovic concerning the relationship of simply connected regions (not the whole plane) to the convex hull boundaries in hyperbolic 3-space that face them. Vlad's talk in turn will discuss advanced aspects of the theory and some of the latest results.


V. Markovic (Warwick) Title: Geometry of simply connected domains and the convex hull boundary

Abstract: In this talk I will present some recent results concerning the relation between simply connected domains and the corresponding convex hull boundaries. Some results related to the geometry of simply connected domains (in particular domains invariant under the action of some finitely generated Kleinian group) will be discussed as well. These results are obtained in joint work with David Epstein and Al Marden.


A. Mednykh (Novosibirsk) Title: On hyperbolic and spherical volumes for knot and link orbifolds

Abstract: Orbifold and cone-manifold structures on two-bridge knots and links are investigated. We show how to construct hyperbolic, Euclidean, and spherical structures on a family of two-bridge cone-manifolds. Isoperimetric inequalities relating cone-manifold volumes and lengths of singular geodesics are obtained . For some cases explicit formulae for volumes and lengths are given.


H. Miyachi (Osaka City) Title: Geometry of one dimensional Teichmueller space

Abstract: In this talk, I will discuss geometric properties of the images of complex analytic embeddings of one dimensional Teichmueller space. I will focus on the Bers slice, the Earle slice, and the Maskit slice.


J.P. Otal (Orleans)


J. Parker (Durham) Title: Geometrically infinite punctured torus groups

Abstract: In a well known unpublished article, Jorgensen used his method of geometrical continuity to develop a structure theorem for the Ford domains of quasi-Fuchsian punctured torus groups.Using Minsky's theorem about the closure of quasi-Fuchsian punctured torus space, following Akiyoshi we can extend Jorgensen's theorem to geometrically infinite punctured torus groups. The key parameter here is the end-invariant. Using Jorgensen's result we produce a tetrahedral decomposition of such manifolds. This enables us to relate the geometry of these manifolds to the diophantine properties of their end-invariants. In particular, the end-invariant is badly approximable if and only if the tetrahedra have bounded shape, equivalently all points of the convex core are a bounded distance from canonical horoballs at the cusp. This clarifies a remark of Sullivan which we turn into a quantitative result.


M.Sakuma (Osaka) Title: Comparing two convex hull constructions for punctured torus groups

Abstract: Let G be a once-punctured torus group, that is, a discrete free subgroup of PSL(2,C) generated by two transformations A and B such that the commutator [A,B] is parabolic. Then the following two convex hulls are associated with G:

(1) The convex hull of the limit set of G in the closure of hyperbolic space. If G is quasifuchsian, then the boundary of the convex hull determines a pair of measured laminations, called the bending laminations.

(2) The convex hull in the Minkowski space of the G-orbit of a light like vector which represents a horoball projecting to the main cusp. Then the boundary of the convex hull projects to a canonical ideal polyhedral decomposition of a subspace of the convex core of the quotient hyperbolic manifold.

It is natural to expect some nice relationship between the two convex hull constructions, and we conjecture that the combinatorial structure of the canonical ideal polyhedral decomposition is determined by the pair of the projective bending laminations, and vice versa. In this talk, we propose a more explicit version of the conjecture, and present a few partial results and computer experiments. This work is based on joint work with H. Akiyoshi, M. Wada, and Y. Yamashita on generalization of Jorgensen's work on the Ford domains of punctured torus groups.


Limited funds will be available to help with expenses for mathematicians based in the UK, especially research students and postdocs.

Supported by the London Mathematical Society

For further details please contact:

Hazel Graley, Mathematics Research Centre,
University of Warwick, Coventry CV4 7AL
hazel@maths.warwick.ac.uk
Tel. 024-76-528317 Fax.024-76-523548