1,262 research outputs found
On the Tits alternative for groups
We prove the Tits alternative for an almost coherent group which is
not virtually properly locally cyclic. In particular, we show that an almost
coherent group which cannot be generated by fewer than four elements
always contains a rank 2 free group.Comment: 16 pages, minor corrections, to appear in Ann. Fac. Sci. Toulouse
Mat
Orderable 3-manifold groups
We investigate the orderability properties of fundamental groups of
3-dimensional manifolds. Many 3-manifold groups support left-invariant
orderings, including all compact P^2-irreducible manifolds with positive first
Betti number. For seven of the eight geometries (excluding hyperbolic) we are
able to characterize which manifolds' groups support a left-invariant or
bi-invariant ordering. We also show that manifolds modelled on these geometries
have virtually bi-orderable groups. The question of virtual orderability of
3-manifold groups in general, and even hyperbolic manifolds, remains open, and
is closely related to conjectures of Waldhausen and others.Comment: 37 pages. Published version. Improvements in the organisation and
presentation of the materia
Reducible And Finite Dehn Fillings
We show that the distance between a finite filling slope and a reducible
filling slope on the boundary of a hyperbolic knot manifold is at most one.Comment: 17 pages, 11 figure
On definite strongly quasipositive links and L-space branched covers
We investigate the problem of characterising the family of strongly
quasipositive links which have definite symmetrised Seifert forms and apply our
results to the problem of determining when such a link can have an L-space
cyclic branched cover. In particular, we show that if is the dual Garside element and is a strongly quasipositive braid whose braid closure is
definite, then implies that is one of the torus links
or pretzel links . Applying
Theorem 1.1 of our previous paper we deduce that if one of the standard cyclic
branched covers of is an L-space, then is one of
these links. We show by example that there are strongly quasipositive braids
whose closures are definite but not one of these torus or pretzel
links. We also determine the family of definite strongly quasipositive
-braids and show that their closures coincide with the family of strongly
quasipositive -braids with an L-space branched cover.Comment: 62 pages, minor revisions, accepted for publication in Adv. Mat
Branched covers of quasipositive links and L-spaces
Let be a oriented link such that , the -fold cyclic cover
of branched over , is an L-space for some . We show that if
either is a strongly quasipositive link other than one with Alexander
polynomial a multiple of , or is a quasipositive
link other than one with Alexander polynomial divisible by , then there is an integer , determined by the Alexander
polynomial of in the first case and the Alexander polynomial of and the
smooth -genus of , , in the second, such that . If
is a strongly quasipositive knot with monic Alexander polynomial such as an
L-space knot, we show that is not an L-space for , and
that the Alexander polynomial of is a non-trivial product of cyclotomic
polynomials if is an L-space for some . Our
results allow us to calculate the smooth and topological 4-ball genera of, for
instance, quasi-alternating quasipositive links. They also allow us to classify
strongly quasipositive alternating links and -strand pretzel links.Comment: 49 pages, 7 figures, minor corrections and improved exposition,
accepted for publication by the Journal of Topolog
On proper powers in free products and Dehn surgery
AbstractWe prove that for any triple (p, q, r) of integers, each greater than or equal to 2, and word w∈Z/p∗Z/q,the group (Z/p∗Z/q)/〈wr〉 is nontrivial. Conditions on the triple (p, q, r) are found under which this quotient group is infinite. We then apply these results to provide the last piece of evidence needed to show that nonintegral Dehn surgery on a knot in S3 cannot yield a reducible 3-manifold
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