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University of Michigan
Department of Mathematics
Winter 2006 Topology Seminar
Mondays 3-4, 3096 East Hall
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For more information, contact Emina Alibegovic.
If you want to check the list of seminars held in previous terms, click on
the appropriate link below.
Fall 2005,
Winter 2005, Fall 2004, Winter 2004, Fall 2003, Winter 2003, Fall 2002, the year 2000-1.
Schedule of Talks
Abstracts
January 30
The central problem in 3-manifold topology is the Homeomorphis
Problem: Given two 3-manifolds M and N to decide if they are
homeomorphic. This is equivalent to the Classification Problem for
3-manifolds. Besides the central interest of a classification of
3-manifolds, the homeomorphism problem for 3-manifolds has the curious
position of being between the beautiful solution for 2-manifolds and
undecidability for n-manifolds when n is larger than 3. The
Homeomorphism Problem for 3-manifolds can be shown to follow from the
Geometrization Conjecture, which has been announced to be true by
G. Perelman. In this lecture, I will outline an algorithm for deciding
if two given 3-manifolds are homeomorphic. The algorithm has a long
journey before it can take advantage of the Geometrization Conjecture;
I will discuss various points along the way where there is opportunity
to improve the algorithm.
February 6
Many important classes of compact manifolds, such as hyperbolic
manifolds, have paltry isometry group but abundant local symmetry. I
will present a structure theorem for compact aspherical Lorentz
manifolds with abundant local symmetry. This result is analogous to a
theorem of Farb and Weinberger on compact aspherical Riemannian
manifolds. Some tools that will be discussed are Gromov's
stratification for rigid geometric structures and rational
cohomological dimension.
February 20
We show that a random walk of length n on the mapping class group
gives a pseudo-Anosov element with a probability that tends to one as
n tends to infinity. As an application we show that if you use the
random walk of length n as the gluing map for a Hegaard splitting, you
get a hyperbolic manifold with a probability that tends to one as n
tends to infinity.
March 27
An outstanding problem in the study of closed 3-manifolds is the
question that how the geometric structures looks like depending on the
topology of the manifold. The major part of the problem is to give a
concrete description of the hyperbolic metric on closed 3-manifolds by
starting from a topological description. We will talk about a new
approach toward this problem which uses the deformation theory of
hyperbolic structures on open manifolds. We will see how this
approach works for a large class of closed 3- manifolds. Furthermore
in order to show the power of this geometric picture we mention how
one can use it to prove many interesting topological facts about these
3-manifolds.
April 3
The following results are proved. (1)The center of any finite index
subgroup of an irreducible, infinite, non-affine Coxeter group is
trivial.
(2)Any finite index subgroup of an irreducible, infinite, non-affine
Coxeter group cannot be expressed as a product of two nontrivial
subgroups.
These two conclusions imply the following.
If a group G is a direct product of n irreducible, infinite,
non-affine Coxeter groups, then any finite index subgroup H of G has a
trivial center, and H can be expressed uniquely as a direct product of
m nontrivial subgroups of H (up to the rearrangement of factors),
where each factor cannot be further decomposed and 1<=m<=n.
Two examples are given to show that m < n can happen.
April 10
We will present a new proof of the Brock-Canary-Minsky Ending
Lamination Theorem. As in the original proof, our starting point will
be Minsky's a priori bounds on lengths of certain closed geodesics in
hyperbolic 3-manifolds homotopy equivalent to a surface. From here the
two proofs diverge. Roughly speaking we avoid many of the difficult
geometric limit arguments of the original proof by using the
techniques of grafting and drilling. This is joint work with
J. Brock, R. Evans and J. Souto.
April 17
Let W be an infinite Coxeter group. Then W acts properly and
cocompactly by isometries on its Davis complex, a finite-dimensional
piecewise-Euclidean CAT(0) complex. On the other hand, in many cases, W
cannot act on a tree without a global fixed point. We study what happens in
the dimensions between 1 and the dimension of the Davis complex of W. In
particular, we consider the minimal dimension of a CAT(0) complex on which W
acts without a global fixed point. We give a complete characterization of
this dimension for many Coxeter groups in terms of a condition on Coxeter
presentations.
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