Michael Brandenbursky

ON QUASI-MORPHISMS FROM KNOT AND BRAID INVARIANTS

We study quasi-morphisms on the groups Pn of pure braids on n strings and on the group D of compactly supported area-preserving diffeomorphisms of an open two-dimensional disc. We show that it is possible to build quasi-morphisms on Pn by using knot invariants which satisfy some special properties. In particular, we study quasi-morphisms which come from knot Floer homology and Khovanov-type homology. We then discuss possible variations of the Gambaudo-Ghys construction, using the above quasi-morphisms on Pn to build quasi-morphisms on the group D of diffeomorphisms of a 2-disc.

On the autonomous metric on the group of area-preserving diffeomorphisms of the 2–disc

EDRA Let D 2 be the open unit disc in the Euclidean plane and let GWD Diff.D 2 ; area/ be the group of smooth compactly supported area-preserving diffeomorphisms of D 2 . For every natural number k we construct an injective homomorphism Z k ! G , which is bi-Lipschitz with respect to the word metric on Z k and the autonomous metric on G . We also show that the space of homogeneous quasimorphisms vanishing on all autonomous diffeomorphisms in the above group is infinite-dimensional. 57S05

Bi-invariant metrics and quasi-morphisms on groups of Hamiltonian diffeomorphisms of surfaces

Let \Sigma_g be a closed orientable surface let Diff_0(\Sigma_g; area) be the identity component of the group of area-preserving diffeomorphisms of \Sigma_g. In this work we present an extension of Gambaudo-Ghys construction to the case of a closed hyperbolic surface \Sigma_g, i.e. we show that every non-trivial homogeneous quasi-morphism on the braid group on n strings of \Sigma_g defines a non-trivial homogeneous quasi-morphism on the group Diff_0(\Sigma_g; area). As a consequence we give another proof of the fact that the space of homogeneous quasi-morphisms on Diff_0(\Sigma_g; area) is infinite dimensional. Let Ham(\Sigma_g) be the group of Hamiltonian diffeomorphisms of \Sigma_g. As an application of the above construction we construct two injective homomorphisms from Z^m to Ham(\Sigma_g), which are bi-Lipschitz with respect to the word metric on Z^m and the autonomous and fragmentation metrics on Ham(\Sigma_g). In addition, we construct a new infinite family of Calabi quasi-morphisms on Ham(\Sigma_g).

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.

QUASI-MORPHISMS AND Lp-METRICS ON GROUPS OF VOLUME-PRESERVING DIFFEOMORPHISMS

Let M be a smooth compact connected oriented manifold of dimension at least two endowed with a volume form μ. We show that every homogeneous quasi-morphism on the identity component Diff0(M, μ) of the group of volume-preserving diffeomorphisms of M, which is induced by a quasi-morphism on the fundamental group π1(M), is Lipschitz with respect to the Lp-metric on Diff0(M, μ). As a consequence, assuming certain conditions on π1(M), we construct bi-Lipschitz embeddings of finite dimensional vector spaces into Diff0(M, μ).

THE CANCELLATION NORM AND THE GEOMETRY OF BI-INVARIANT WORD METRICS

We study bi-invariant word metrics on groups. We provide an efficient algorithm for computing the bi-invariant word norm on a finitely generated free group and we construct an isometric embedding of a locally compact tree into the bi-invariant 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.

On the autonomous norm on the group of Hamiltonian diffeomorphisms of the torus

We prove that the autonomous norm on the group of Hamiltonian diffeomorphisms of the two-dimensional torus is unbounded. We provide explicit examples of Hamiltonian diffeomorphisms with arbitrarily large autonomous norm. For the proofs we construct quasimorphisms on

Quasi-isometric embeddings into diffeomorphism groups

Ham(T^2)

INVARIANTS OF CLOSED BRAIDS VIA COUNTING SURFACES

and some of them are Calabi.

The Lp–diameter of the group of area-preserving diffeomorphisms of S2

Let M be a smooth compact connected oriented manifold of dimension at least two endowed with a volume form. Assuming certain conditions on the fundamental group