 
Summary: RESEARCH BLOG 3/06/03
Elizabeth Klodginski asked whether the injectivity radius of closed 3
manifolds with nonpositively curved (npc) cubings has a lower bound
(n.b.: she has accepted a VRAP at UC Davis for next year). Her moti
vation was to find explicit examples of fibered manifolds which do not
have an npc cubing, since one can find fibered hyperbolic 3manifolds
of arbitrarily small injectivity radius. So a lower bound on the in
jectivity radius of a cubed manifold would give examples of fibered
manifolds that aren't cubed. I believe this question came out of study
ing a construction of Aitchison and Rubinstein of cubed 3manifolds
which are virtually fibered [2]. Liz says that she discovered a problem
with their argument (which is pretty sketchy), but she has shown that
their method works sometimes by working out an explicit example. Re
cently, as part of her thesis, she has also shown that surfaces in fibered
hyperbolic 3manifolds which are transverse to the pseudoAnosov flow
do not satisfy the 1line condition (so in particular, they could not be
the canonical 1injective surface associated to a cubing), answering a
question of Alan Reid. She gave a talk at UIC on this work, which
makes nice use of the dynamics of the pA map and work of Cooper,
Long, and Reid on surfaces transverse to pA flows of fibered manifolds
