24 research outputs found
Concordance group and stable commutator length in braid groups
We define a quasihomomorphism from braid groups to the concordance group of
knots and examine its properties and consequences of its existence. In
particular, we provide a relation between the stable four ball genus in the
concordance group and the stable commutator length in braid groups, and produce
examples of infinite families of concordance classes of knots with uniformly
bounded four ball genus. We also provide applications to the geometry of the
infinite braid group. In particular, we show that its commutator subgroup
admits a stably unbounded conjugation invariant norm. This answers an open
problem posed by Burago, Ivanov and Polterovich.Comment: 25 pages, 5 figure
On Lipschitz functions on groups equipped with conjugation-invariant norms
We observe that a function on a group equipped with a bi-invariant word
metric is Lipschitz if and only if it is a partial quasimorphism bounded on the
generating set. We also show that an undistorted element is always detected by
a homogeneous partial quasimorphisms. We provide a general homogenisation
procedure for Lipschitz functions and relate partial quasimorphisms on a group
to ones on its asymptotic cones.Comment: 11 page
Pseudoholomorphic tori in the Kodaira-Thurston manifold
The Kodaira-Thurston manifold is a quotient of a nilpotent Lie group by a
cocompact lattice. We compute the family Gromov-Witten invariants which count
pseudoholomorphic tori in the Kodaira-Thurston manifold. For a fixed symplectic
form the Gromov-Witten invariant is trivial so we consider the twistor family
of left-invariant symplectic forms which are orthogonal for some fixed metric
on the Lie algebra. This family defines a loop in the space of symplectic
forms. This is the first example of a genus one family Gromov-Witten
computation for a non-K\"ahler manifold.Comment: 46 pages; v2 added some references and explanation, v3 couple of
typos corrected. To appear in Compositio Mathematic
Cancelation norm and the geometry of biinvariant word metrics
We study biinvariant word metrics on groups. We provide an efficient
algorithm for computing the biinvariant word norm on a finitely generated free
group and we construct an isometric embedding of a locally compact tree into
the biinvariant Cayley graph of a nonabelian free group. We investigate the
geometry of cyclic subgroups. We observe that in many classes of groups cyclic
subgroups are either bounded or detected by homogeneous quasimorphisms. We call
this property the bq-dichotomy and we prove it for many classes of groups of
geometric origin.Comment: 32 pages, to appear in Glasgow Journal of Mathematic
Strong and uniform boundedness of groups
A group G is called bounded if every conjugation-invariant norm on G has
finite diameter. We introduce various strengthenings of this property and
investigate them in several classes of groups including semisimple Lie groups,
arithmetic groups and linear algebraic groups. We provide applications to
Hamiltonian dynamics.Comment: 37 pages; preliminary version; second version substantially revise