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Given a compact orientable surface of negative Euler characteristic, there exists a natural pairing between the Teichmüller space of the surface and the set of homotopy classes of simple loops and arcs. The length pairing sends a hyperbolic metric and a homotopy class of a simple loop or arc to the length of geodesic in its homotopy class. We study this pairing function using the Fenchel–Nielsen coordinates on Teichmüller space and the Dehn–Thurston coordinates on the space of homotopy classes of curve systems. Our main result establishes Lipschitz type estimates for the length pairing expressed in terms of these coordinates. As a consequence, we reestablish a result of Thurston–Bonahon that the length pairing extends to a continuous map from the product of the Teichmüller space and the space of measured laminations.
The Jones–Witten theory gives rise to representations of the (extended) mapping class group of any closed surface indexed by a semi-simple Lie group and a level . In the case these representations (denoted ) have a particularly simple description in terms of the Kauffman skein modules with parameter a primitive th root of unity (). In each of these representations (as well as the general case), Dehn twists act as transformations of finite order, so none represents the mapping class group faithfully. However, taken together, the quantum representations are faithful on non-central elements of . (Note that has non-trivial center only if is a sphere with or punctures, a torus with or punctures, or the closed surface of genus .) Specifically, for a non-central there is an such that if and is a primitive th root of unity then acts projectively nontrivially on . Jones’ original representation of the braid groups , sometimes called the generic –analog––representation, is not known to be faithful. However, we show that any braid admits a cabling so that , .
We show that if is a surface bundle over with fiber of genus 2, then for any integer , has a finite cover with . A corollary is that can be geometrized using only the “non-fiber" case of Thurston’s Geometrization Theorem for Haken manifolds.
The theory of Vassiliev invariants deals with many modules of diagrams on which the algebra defined by Pierre Vogel acts. By specifying a quadratic simple Lie superalgebra, one obtains a character on . We show the coherence of these characters by building a map of graded algebras beetwen and a quotient of a ring of polynomials in three variables; all the characters induced by simple Lie superalgebras factor through this map. In particular, we show that the characters for the Lie superalgebra with dimension 40 and for are the same.
Let be an orientable and irreducible –manifold whose boundary is an incompressible torus. Suppose that does not contain any closed nonperipheral embedded incompressible surfaces. We will show in this paper that the immersed surfaces in with the –plane property can realize only finitely many boundary slopes. Moreover, we will show that only finitely many Dehn fillings of can yield –manifolds with nonpositive cubings. This gives the first examples of hyperbolic –manifolds that cannot admit any nonpositive cubings.
We give sufficient conditions for a group of homeomorphisms of a Peano continuum without cut-points to be a convergence group. The condition is that there is a collection of convergence subgroups whose limit sets “cut up" in the correct fashion. This is closely related to the result in [Topology 39 (2000) 229-237].
This article describes various moduli spaces of pseudoholomorphic curves on the symplectization of a particular overtwisted contact structure on . This contact structure appears when one considers a closed self dual form on a 4–manifold as a symplectic form on the complement of its zero locus. The article is focussed mainly on disks, cylinders and three-holed spheres, but it also supplies groundwork for a description of moduli spaces of curves with more punctures and non-zero genus.
Let be a three-dimensional –orbifold, with branching locus a knot transverse to the Seifert fibration. We prove that is the limit of hyperbolic cone manifolds with cone angle in . We also study the space of Dehn filling parameters of . Surprisingly it is not diffeomorphic to the deformation space constructed from the variety of representations of . As a corollary of this, we find examples of spherical cone manifolds with singular set a knot that are not locally rigid. Those examples have large cone angles.
We formulate and prove a chain rule for the derivative, in the sense of Goodwillie, of compositions of weak homotopy functors from simplicial sets to simplicial sets. The derivative spectrum of such a functor at a simplicial set can be equipped with a right action by the loop group of its domain , and a free left action by the loop group of its codomain . The derivative spectrum of a composite of such functors is then stably equivalent to the balanced smash product of the derivatives and , with respect to the two actions of the loop group of . As an application we provide a non-manifold computation of the derivative of the functor .
The main theorem of this paper is a generalisation of well known results about Dehn surgery to the case of attaching handlebodies to a simple 3–manifold. The existence of a finite set of ‘exceptional’ curves on the boundary of the 3–manifold is established. Provided none of these curves is attached to the boundary of a disc in a handlebody, the resulting manifold is shown to be word hyperbolic and ‘hyperbolike’. We then give constructions of gluing maps satisfying this condition. These take the form of an arbitrary gluing map composed with powers of a suitable homeomorphism of the boundary of the handlebodies.
Given a hyperbolic 3–manifold containing an embedded closed geodesic, we estimate the volume of a complete hyperbolic metric on the complement of the geodesic in terms of the geometry of . As a corollary, we show that the smallest volume orientable hyperbolic 3–manifold has volume .
A –local finite group is an algebraic structure with a classifying space which has many of the properties of –completed classifying spaces of finite groups. In this paper, we construct a family of 2–local finite groups, which are exotic in the following sense: they are based on certain fusion systems over the Sylow 2–subgroup of ( an odd prime power) shown by Solomon not to occur as the 2–fusion in any actual finite group. Thus, the resulting classifying spaces are not homotopy equivalent to the –completed classifying space of any finite group. As predicted by Benson, these classifying spaces are also very closely related to the Dwyer–Wilkerson space .