Spring Lecture Series 2003 Contributed Talks

• Speaker:  Ian Agol
  Title:  Marden's conjecture and exceptional Dehn fillings
  Abstract:
  Marden's conjecture states that a complete hyperbolic 3-manifold with finitely generated fundamental group is tame, i.e. the interior of a compact 3-manifold. Given a manifold with torus boundary and hyperbolic interior (a hyperbolic knot complement), an exceptional Dehn filling is one which is reducible, or has finite fundamental group, or non-word-hyperbolic fundamental group. Assuming Marden's conjecture, we show that there are only finitely many 1-cusped hyperbolic knot complements which have > 8 exceptional Dehn fillings. It is conjectured that there are only finitely many with > 6 exceptional fillings, and an explicit list of such manifolds is conjectured to be the only examples. We will also show that for a non-compact hyperbolic manifold M with b_1(M)>2, the volume of M is >= 2pi*v_3/sqrt(3), where v_3 is the volume of a regular ideal tetrahedron in H^3. The common thread of these two theorems is a result of Anderson, Canary, Culler and Shalen which gives an improved Margulis lemma for tame free groups.




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• Speaker:  Mark Brittenham
  Title:  Knots with unique minimal genus Seifert surface
  Abstract:
  We show how to build families of knots with unique minimal genus Seifert surfaces, and apply this to construct hyperbolic knots with depth greater than one.




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• Speaker:  Sergio Fenley
  Title:  3-manifolds, laminations and group actions
  Abstract:
  Manifolds with essential laminations have fantastic properties, for example they are irreducible and have universal cover homeomorphic to R^3. Suppose that a 3-manifold M has an essential lamination L. If there are leaves isolated on both sides, blow each one to an I-bundle of leaves. Then in the universal cover of M, the leaf space of the lifted lamination is an example of a non Hausdorff tree and the fundamental group acts on it. After a further modification one produces a non trivial action on an actual tree. Some specific group actions on trees coming from manifolds obtained by Dehn surgery on torus bundles over the circle can be analysed in detail. These give information about the existence problem for essential laminations in 3-manifolds.




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• Speaker:  Hiroshi Goda
  Title:  Heegaard splitting for sutured manifolds and Circle valued Morse theory for knots and links
  Abstract:
  We show Circle valued Morse theory for knots and links in the (homology) 3-sphere. A handle decomposition corresponding to a circle valued Morse function can be regarded as Heegaard splitting for sutured manifold. We discuss behaviors of the Heegaard genera and its estimate using knot invariants.




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• Speaker:  Shelly Harvey
  Title:  Some remarks on the Virtual Betti Number of a 3-manifold
  Abstract:
  It is conjectured that every closed 3-manifold M has finite covers with arbitrarily large first betti number. We discuss this question for arbitrary abelian covers. In particular, we give a sequence of algebraic and topological properties which guarantee the existence of a cyclic cover with "large" first betti number and discuss some particular examples.




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• Speaker:  Ben Klaff
  Title:  Boundary slopes of knots in closed 3-manifolds with cyclic fundamental group
  Abstract:
  We show that if N is a closed 3-manifold with odd cyclic fundamental group and K is a (tame) knot in N such that the exterior of K is irreducible, then at least one of the following holds: (1) for every framing of the knot K, there exists a boundary slope whose absolute value is greater than one; or (2) K is an iterated cable of a knot whose exterior is a solid torus. The main argument in the proof, based on N. Dunfield�s argument in the case where K is hyperbolic, brings to bear some deep results in the theory of character varieties of hyperbolic 3-manifolds.




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• Speaker:  Jason Manning
  Title:  Quasi-actions on trees and the geometry of pseudocharacters
  Abstract:
  We show how a nontrivial pseudocharacter on a group gives rise to a quasi-action on a tree. This gives rise to examples of exotic quasi-actions on bushy (infinite valence) trees. We also give some examples of groups which cannot quasiact coboundedly on any infinite tree.
  
  http://front.math.ucdavis.edu/math.GR/0303380




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• Speaker:  Roman Mikhailov
  Title:  Augmentation powers, group homologies and localizations
  Abstract:
  We consider generalized polinomial filtrations in group cohomologies and Dwyer's filtrations (which play important role in the 4-dimensional topology), consider transfinite conditions on this filtrations. Also we define transfinite parafree groups and study some "homologically wild" groups.(This is a joint work with I.B.S.Passi)




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• Speaker:  Jennifer Schultens
  Title:  Some topology and algebra of graph manifolds
  Abstract:
  Graph manifolds possess enough structure to allow for many types of calculations. Under certain orientability assumptions, enough is known about the structure of the Heegaard splittings of a graph manifold to determine its Heegaard genus. Joint work with Richard Weidmann shows that the difference between the Heegaard genus and the rank of the fundamental group of a graph manifold can be arbitrarily large.