Summary: RESEARCH BLOG 6/14/04
TAMENESS IN TEXAS
Two weekends ago, Jeff Brock organized a conference Tameness in
Texas, which was centered around the classification of Kleinian groups.
I have a preprint available, which shows that every hyperbolic mani-
fold with finitely generated fundamental group is homeomorphic to the
interior of a compact manifold with boundary (so the ends are home-
omorphic to a closed surface ŚR), answering the Marden conjecture
(also called tameness), see blog 11/12/03 . As it happened, Calegari
and Gabai have a proof as well, and Gabai gave talks at UT Austin the
Thursday before the conference, as well as at the conference. In fact,
Suhyoung Choi also has an argument, making use of one idea outlined
in blog 11/14/03, but otherwise independent of my proof. In fact, all
three arguments have very similar structure and ideas. Calegari and
Gabai's proof is remarkable, in that they make no use of Bonahon
or Canary's previous work on tameness in the indecomposable case.
Their argument makes use of a notion of "shrinkwrap", which is a
surface which is made minimal with geodesic barriers, and has an in-
trinsic CAT(-1) structure. This replaces pleated surfaces introduced
by Thurston or simplicial surfaces introduced by Bonahon and Canary.