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RESEARCH BLOG 4/22/03 Last Wednesday, Howie Masur gave a talk on the curve complex of

Summary: RESEARCH BLOG 4/22/03
Last Wednesday, Howie Masur gave a talk on the curve complex of
handlebodies. He and Yair Minsky have answered a question which
I wondered about when I first saw their papers on the hyperbolicity
of the curve complex (see [1]). Given a surface S (we'll assume it's
closed, for simplicity), the 1-skeleton of the curve complex C(S) has
vertices consisting of isotopy classes of simple closed curves on the
surface, and edges between curves which are disjoint (and distinct).
If S = H is the boundary of a handlebody, then there is a subset
C(H) C(S) consisting of the subcomplex spanned by curves in S
which bound disks in H. What Masur and Minsky proved is that
C(H) is quasiconvex in C(S), that is every geodesic in C(S) connecting
vertices of C(H) is within a bounded distance K = K(S) of C(H) [2].
For each pair of vertices c1, c2 C(H), they find a path in C(H)
connecting c1 and c2 which is a K quasi-convex subset of C(S), from
which the theorem follows. Since c1 and c2 bound disks in H, the usual
outermost disk argument shows that c1 will have a wave with respect
to c2. One does surgery along this wave to get a curve c disjoint
from c2 which has fewer intersections with c1 and still bounds a disk
in H, so is a vertex of C(H). Continuing carefully in this fashion,


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


Collections: Mathematics