 
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 1skeleton 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 quasiconvex 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,
