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RESEARCH BLOG 11/14/03 I claimed last time that I wanted to reduce the general case of Mar-

Summary: RESEARCH BLOG 11/14/03
I claimed last time that I wanted to reduce the general case of Mar-
den's conjecture to the special case when M - g may be exhausted by
incompressible surfaces. I don't know how to do this, but I think one
may be able to get around some difficulties by implementing techniques
of Canary, Minsky, and Souto directly.
First, I need to recap some of the ideas of Bob Myers adapted to the
context I set up in the last blog 11/13/03. A year ago, I came up with
some similar ideas by thinking about minimal surface representatives
of incompressible surfaces, but Myers' approach is simpler, and takes
advantage of technology already in the literature. What Myers proves
is that there is an open submanifold g V M, such that V is an
end reduction of M at g (this notion appears to be due to Brin and
Thickstun, who prove that it is unique up to non-ambient isotopy).
This means that V - g is irreducible, M - V has no components
with compact closure, V has an exhaustion by compact submanifolds
containing g whose boundary is incompressible in V - g , and any
incompressible surface in M - g can be isotoped in M - g to lie in
V - g (so V engulfs all the incompressible surfaces in M - g ). The
relevant fact that Myers shows is that 1(V ) = 1(M). It is also clear


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


Collections: Mathematics