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RESEARCH BLOG 3/06/03 Elizabeth Klodginski asked whether the injectivity radius of closed 3-

Summary: RESEARCH BLOG 3/06/03
Elizabeth Klodginski asked whether the injectivity radius of closed 3-
manifolds with non-positively 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 3-manifolds
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 3-manifolds
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 3-manifolds which are transverse to the pseudo-Anosov flow
do not satisfy the 1-line condition (so in particular, they could not be
the canonical 1-injective 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


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


Collections: Mathematics