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RESEARCH BLOG 11/13/03 The approach I'm taking to Marden's conjecture is inspired by work
 

Summary: RESEARCH BLOG 11/13/03
The approach I'm taking to Marden's conjecture is inspired by work
of Freedman, which I learned during his seminar as a graduate student.
Let < PSL(2, C) be the group, and M = H3
/ be the manifold
quotient (for simplicity, assume that has no parabolic elements). We
may assume that is decomposable, since the indecomposable case
was taken care of by Bonahon [2]. For simplicity, assume that is
free. Choose an algebraically diskbusting element in , that is an
element which is not homotopic into any free factor of (for example,
the product of the squares of generators of , which is the relator for a
closed non-orientable surface, so we know it cannot be in any free factor
of ). We may take the unique geodesic representative g M for .
Lemma 5.5 of Canary [3] implies that we can perturb the metric on M
to get a negatively curved metric, such that we have a representative g
for which is embedded, and has a small tubular neighborhood which is
locally isometric to H3
. Then we may drill out a tubular neighborhood
of g and fill back in a complete negatively curved metric (this trick is
essentially due to Gromov and Thurston, although it is simpler than

  

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

 

Collections: Mathematics