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RESEARCH BLOG 3/27/04, STDC '04 I went to two days of the Spring Topology and Dynamics Conference,
 

Summary: RESEARCH BLOG 3/27/04, STDC '04
I went to two days of the Spring Topology and Dynamics Conference,
held at University of Alabama at Birmingham. Jim Cannon started off
with a talk on subdivision tilings for hyperbolic groups. For a long
time, he and Floyd and Parry have been working on the conjecture
that irreducible closed 3-manifolds with infinite word-hyperbolic fun-
damental group are actually hyperbolic. This would be one step in an
approach to the geometrization conjecture, which I think would still
be quite interesting program to carry through, even if Perelman's ar-
guments using Ricci flow are correct.
Cannon discussed joint work with Floyd, Hersonsky, and Parry, in
which they show that to a hyperbolic group , one may associate a
subdivision tiling which in some sense recursively constructs . The
main example of such subdivision tilings known before was for compact
NPC cubed spaces. In Cannon's classic paper [1], he shows that for
a hyperbolic manifold, there are only finitely many "cone types". A
cone type is a subset of the Cayley graph with respect to a fixed set
of generators, which (roughly) is the set of all geodesics which have a
common prefix (up to equivalence by truncating prefixes). Cannon's
argument generalizes directly to hyperbolic groups. Take the ball of

  

Source: Agol, Ian - Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago

 

Collections: Mathematics