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RESEARCH BLOG 8/03/04 Last week, I. Kapovich and Weidmann posted a paper showing that

Summary: RESEARCH BLOG 8/03/04
Last week, I. Kapovich and Weidmann posted a paper showing that
there is an algorithm to solve the rank problem for torsion free, convex
cocompact Kleinian groups. I haven't read enough of their paper to
understand how it goes, but from what I can gather, they use gener-
alizations of Nielsen techniques to actions on word-hyperbolic spaces,
which they have developed in previous papers. They also make use
of tameness of subgroups. It would be interesting to generalize their
result to show that there is an algorithm to compute the rank for all
fundamental groups of (good) compact 3-orbifolds.
Boileau and Weidmann have a paper which gives a structure theorem
for 3-manifolds with non-trivial JSJ decomposition and which have two-
generated fundamental group. They would like to rule out one case,
which comes down to showing that a 2-bridge link cannot be an irregu-
lar cover of a manifold with a single boundary torus. I believe it should
be possible to show this using the techniques I've developed for classi-
fying 2-parabolic generator groups. In fact, I think one should be able
to classify the commensurators of (non-arithmetic) 2-bridge links, and
then show that there is no manifold cover between a 2-bridge link and
the minimal orbifold which it covers (one may deal with the arithmetic


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


Collections: Mathematics